CHAPTER Integers and Number Theory - University of...

5Integers and Number Theory

C H A P T E R 5

Preliminary ProblemFind a quick way to compute the sum without using a calculator.

502 - 492 + 482 - 472 + Á + 22 - 12

248

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A Problem Solving Approach to Mathematics for Elementary School Teachers, Tenth Edition, by Rick Billstein, Shlomo Libeskind, and Johnny W. Lott. Published by Addison-Wesley. Copyright © 2010 by Pearson Education, Inc.

Integers and Number Theory 249

▲▲HistoricalNote

The Hindu mathematician Brahmagupta (ca. 598–665 CE) provided the first systematictreatment of negative numbers and of zero. Only about 1000 years later did the Italianmathematician Gerolamo Cardano (1501–1576) consider negative solutions to certainequations. Still uncomfortable with the concept of negative numbers, he called them“fictitious” numbers. ▲▲

he Principles and Standards expectations for students in grades 3–5 include thefollowing:

• explore numbers less than 0 by extending the number line and through familiar applications;

• describe classes of numbers according to characteristics such as the nature of their factors. (p. 148)

The expectations for students in grades 6–8 include:

• use factors, multiples, prime factorization, and relatively prime numbers to solve problems;

• develop meaning for integers and represent and compare quantities with them. (p. 214)

In addition, the Principles and Standards points out that in grades 6–8:

Students can also work with whole numbers in their study of number theory. Tasks, such as thefollowing, involving factors, multiples, prime numbers, and divisibility can afford opportunities forproblem solving and reasoning.

1. Explain why the sum of the digits of any multiple of 3 is itself divisible by 3.

2. A number of the form abcabc always has several prime-number factors. Which prime num-bers are always factors of a number of this form? Why?

Middle-grades students should also work with integers. In lower grades, students may haveconnected negative integers in appropriate ways to informal knowledge derived from everyday experi-ences, such as below-zero winter temperatures or lost yards on football plays. In the middle grades,students should extend these initial understandings of integers. Positive and negative integers shouldbe seen as useful for noting relative changes or values. Students can also appreciate the utility of nega-tive integers when they work with equations whose solution requires them, such as (pp.217–18)

In this chapter, we start with the system of integers and then develop an understanding ofnumber theory.

Negative numbers are useful in everyday life. For example, Mount Everest is 29,028 ftabove sea level, and the Dead Sea is 1293 ft below sea level. We may symbolize these eleva-tions as 29,028 and

In mathematics, the need for negative integers arises because subtractions cannot alwaysbe performed in the set of whole numbers. To compute using the definition of sub-traction for whole numbers, we must find a whole number n such that There isno such whole number n. To perform the computation, we must invent a new number, a

6 + n = 4.4 - 6

-1293.

2x + 7 = 1.

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250 Integers and Number Theory

5-1

Other numbers to the left of 0 are created similarly. The new set of numbersis the set of negative integers. The set is the set of

positive integers. The integer 0 is neither positive nor negative. The union of the set ofnegative integers, the set of positive integers, and is the set of integers, denoted by I.

As a field of study, number theory started to flourish in the seventeenth century with thework of Pierre de Fermat (1601–1665). Topics in number theory that occur in the elemen-tary school curriculum include factors, multiples, divisibility tests, prime numbers, primefactorizations, greatest common divisors, and least common multiples. The topic of con-gruences, introduced by Carl Gauss (1777–1855), is also incorporated into the elementarycurriculum through clock arithmetic and modular arithmetic. Clock and modular arithmeticgive students a look at a mathematical system.

I = 5Á ,-4, -3, -2, -1, 0, 1, 2, 3, 4, Á 6

506

51, 2, 3, 4, Á 65-1, -2, -3, -4, Á 6

–5

4 – 6

–6

0 1 2 3 4 5 6–4 –3 –2 –1

4

Figure 5-1

negative integer. If we attempt to calculate on a number line, then we must drawintervals to the left of 0. In Figure 5-1, is pictured as an arrow that starts at 0 andends 2 units to the left of 0. The new number that corresponds to a point 2 units to the leftof 0 is negative two, symbolized by -2.

4 - 64 - 6


The dash has not always been used for both the subtraction operation and the negativesign. Other notations were developed but never adopted universally. One such notation wasused by Abu al-Khwârizmî (ca. 825), who indicated a negative number by placing a smallcircle over it. For example, was recorded as The Hindus denoted a negative numberby enclosing it in a circle; for example, was recorded as ➃ . The symbols and firstappeared in print in European mathematics in the late fifteenth century, at which time thesymbols referred not to addition or subtraction nor positive or negative numbers, but tosurpluses and deficits in business problems.

-+-44° .-4

▲▲

Integers and the Operations of Addition and Subtraction

Representations of IntegersIt is unfortunate that we use the symbol “ ” to indicate both a subtraction and a negative sign.To reduce confusion between the uses of this symbol in this text, a raised “ ” sign is used fornegative numbers, as in and for the opposite of a number, as in in contrast to the lowersign for subtraction. To emphasize that an integer is positive, sometimes a raised plus sign isused, as in In this text, we use the plus sign for addition only and write simply as 3.+3+3.

-x,-2,

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Section 5-1 Integers and the Operations of Addition and Subtraction 251

The value of in Example 5-1(b) is 5. Note that is the opposite of x and might not repre-sent a negative number. In other words, x is a variable that can be replaced by some numbereither positive, zero, or negative. Note: is read “the opposite of x” not “minus x” or “negative x.”

In the grade 7 Focal Points we find the following:

By applying properties of arithmetic and considering negative numbers in everyday contexts (e.g.,situations of owing money or measuring elevations above and below sea level), students explain whythe rules for adding, subtracting, multiplying, and dividing with negative numbers make sense. (p. 19)

We next investigate many informal ways to introduce operations on integers; we first beginby looking at addition of integers.

Integer AdditionAs mentioned in the Research Note, it is important to use hands-on materials whenworking with integers. Several models are presented next to motivate integer addition.Teachers can actually build a number line for students to walk when using the number-line model.

Chip Model for Addition

In the chip model, positive integers are represented by black chips and negative integers byred chips. One red chip neutralizes one black chip. Hence, the integer can be repre-sented by 1 red chip, or 2 red and 1 black, or 3 red and 2 black, and so on. Similarly, everyinteger can be represented in many ways using chips, as shown in Figure 5-2.

-1

-x

-x-x

Research NoteIt is important to usemanipulatives whenworking with nega-tive numbers(Thompson 1988). ▲▲

▲▲

E-ManipulativeActivity

Use the Token Additionactivity for additionalpractice using the chipor charged-fieldmodels.

For each of the following, find the opposite of x:

a.b.c.

Solution a.b.c. -x = -0 = 0

-x = - (-5) = 5

-x = -3

x = 0x = -5x = 3▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-1

▲

▲

R E M A R K Using addition of integers, we shall soon see that when an opposite of aninteger is added to the integer the sum is 0. In fact, can be defined as the solution ofx + a = 0.

-a

The negative integers are opposites of the positive integers. For example, the opposite of5 is Similarly, the positive integers are the opposites of the negative integers. Becausethe opposite of 4 is denoted the opposite of can be denoted which equals 4.The opposite of 0 is 0. In the set of integers I, every element has an opposite that is also in I.

-1-42,-4-4,

-5.

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Figure 5-3 shows a chip model for the addition We put 4 red chips together with3 black chips. Because 3 red chips neutralize 3 black ones, Figure 5-3 represents the equiva-lent of 1 red chip or

Charged-Field Model for Addition

A model similar to the chip model uses positive and negative charges. A field has 0 charge ifit has the same number of positive and negative charges. As in the chip model, agiven integer can be represented in many ways using the charged-field model. Figure 5-4uses the model for Because 3 positive charges “neutralize” 3 negative charges, thenet result is 2 negative ones. Hence,

Number-Line ModelAnother model for addition of integers involves a number line, and it can be introducedwith the idea of a hiker walking the number line, as seen on the student page on page 253.Study the student page to see how the model for the hiker works and then how the hiker’smoves might be recorded on a number line. Without the hiker, can be picturedas in Figure 5-5.

-3 + -5

3 + -5 = -2.3 + -5.

1-21+2

-1.

-4 + 3.

+ + +– – – – –

3 + – 5 = +2

– 4 + 3 = –1

Figure 5-3

Figure 5-4

–5

–5

0 1 2 3 4 5 6–4 –3 –2 –1–6–7–8 7 8

–3

–3 + –5

Figure 5-5


Use the Number Lineactivity to illustrate thenumber-line model.

+2

–2 –2 –2 –2

+2 +2 +2

Figure 5-2

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Source: Scott Foresman-Addison Wesley, Grade 6, 2008 (p. 418).

School Book Page ADDING INTEGERS

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Pattern Model for Addition

Addition of whole numbers was discussed in Chapter 3. Addition of integers can also bemotivated by using patterns of addition of whole numbers. Notice that in the left columnof the following list, the first four facts are known from whole-number addition. Also no-tice that the 4 stays fixed and as the numbers added to 4 decrease by 1, the sum decreasesby 1. Following this pattern, and we can complete the remainder of the firstcolumn. Similar reasoning can be used to complete the computations in the right column,where stays fixed and the other numbers decrease by 1 each time.

Note that the reasoning with patterns is inductive reasoning and therefore does not consti-tute a proof.

4 + -6 = -2 -2 + -5 = -7 4 + -5 = -1 -2 + -4 = -6 4 + -4 = 0 -2 + -3 = -5 4 + -3 = 1 -2 + -2 = -4 4 + -2 = 2 -2 + -1 = -3 4 + -1 = 3 -2 + 0 = -2

4 + 0 = 4 -2 + 1 = -1 4 + 1 = 5 -2 + 2 = 0 4 + 2 = 6 -2 + 3 = 1 4 + 3 = 7 -2 + 4 = 2

-2

4 + -1 = 3–5˚–4˚–3˚–2˚–1˚

0˚1˚2˚3˚4˚5˚6˚

+10˚

Figure 5-7

Example 5-2 involves a thermometer with a scale in the form of a vertical number line.

The temperature was In an hour, it rose What is the new temperature?

Solution Figure 5-7 shows that the new temperature is and that -4 + 10 = 6.6°C

10°C.-4°C.

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-2

▲

▲

–5

3 + –5

0 1 2 3 4 5 6–4 –3 –2 –1–6–7–8 7 8

3

–5

Figure 5-6

Figure 5-6 similarly depicts integer addition of 3 + -5.

NOW TRY THIS 5-1

a. Refer to Example B on the student page. Is the sum of two negative integers always negative?b. Refer to Examples A and C on the student page. Is the sum of a positive and a negative integer

positive or negative? Explain.c. Use a number line to add .6 + (-8) + (-2)

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Evaluate each of the following:

a.b.c.d.e.

Solution a.b.c.d.e. ƒ 2 + -5 ƒ = ƒ -3 ƒ = 3

- ƒ -3 ƒ = -3ƒ 0 ƒ = 0ƒ -5 ƒ = 5ƒ 20 ƒ = 20

ƒ 2 + -5 ƒ- ƒ -3 ƒƒ 0 ƒƒ -5 ƒƒ 20 ƒ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-3▲

▲

–5 0 1 2 3 4 5–4 –3 –2 –1

4 units 4 units

Figure 5-8

Distance is always a positive number or zero. The distance between the point corre-sponding to an integer and 0 is the absolute value of the integer. Thus, the absolute valueof both 4 and is 4, written , respectively. Notice that if then and if then is positive. Therefore, we have the following:-xx 6 0,ƒ x ƒ = x,

x Ú 0,ƒ 4 ƒ = 4 and ƒ -4 ƒ = 4-4

Definition of Absolute Value

ƒ x ƒ = -x if x 6 0 ƒ x ƒ = x if x Ú 0

R E M A R K Some students want to shorten the above definition to This is nottrue, as has only one value.ƒ x ƒ

ƒ x ƒ = ; x.

Absolute ValueBecause 4 and are opposites, they are on opposite sides of 0 on the number line and arethe same distance (4 units) from 0, as shown in Figure 5-8.

-4

TECHNOLOGY CORNER On a spreadsheet, in column A enter 4 and fill down 20 rows. (For help onspreadsheets, see the Technology Manual.) In column B, enter 3 as the first entry and then write a formulato add to 3 for the second entry, add to the second entry to get the third entry, and fill down con-tinuing the pattern. In column C, find the sum of the respective entries in columns A and B. What patternsdo you observe? Repeat the problem by changing the entries in column A to and repeating the process.-4

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R E M A R K Notice that by definition the additive inverse, is the solution of theequation The fact that the additive inverse is unique is equivalent to sayingthat the preceding equation has only one solution. In fact, for any integers a and b, theequation has a unique solution, b + -a.x + a = b

x + a = 0.

-a,

The uniqueness of additive inverses can be used to justify other theorems. For example,the opposite, or the additive inverse, of can be written However, because-1-a2.-a

Properties of Integer AdditionInteger addition has all the properties of whole-number addition. These properties can beproved if addition of integers is defined in terms of whole number addition and subtraction.

Theorem 5–2: Uniqueness of the Additive InverseFor every integer a, there exists a unique integer the additive inverse of a, such thata + -a = 0 = -a + a.

-a,

Notice the name identity element in Theorem 5–1. Zero is the identity element of additionbecause when it is added to any integer it does not change the result; it leaves the integerunchanged.

We have seen that every integer has an opposite. This opposite is the additive inverse ofthe integer. The fact that each integer has a unique (one and only one) additive inverse isstated in Theorem 5–2.

Theorem 5–1: Properties

Given integers a, b, and c:

Closure property of addition of integers is a unique integer.

Commutative property of addition of integers .

Associative property of addition of integers

Identity element of addition of integers 0 is the unique integer such that, for allintegers a, 0 + a = a = a + 0.

1a + b2 + c = a + 1b + c2.

a + b = b + a

a + b

NOW TRY THIS 5-2 Write each of the following in simplest form without the absolute value notationin the final answer. Show your work.

a. if b. if c. if x Ú 0- ƒ x ƒ + x

x … 0- ƒ x ƒ + xx … 0ƒ x ƒ + x

R E M A R K It is possible to describe addition of integers as the process of finding thedifference or the sum of the absolute values of the integers and attaching anappropriate sign.

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Find the additive inverse of each of the following:

a.b.c. -3 + -x

a + -4

-13 + x2▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-4

We prove the second part of Theorem 5–3 as follows: By definition is the addi-tive inverse of , that is, If we could show that isalso the additive inverse of , the uniqueness of the additive inverse implies that

and are equal. To show that is also the additive inverse of, we need only to show that This can be shown using the

associative and commutative properties of integer addition and the definition of the additiveinverse as follows:

Now we have

Hence, -1a + b2 = -a + -b.

1a + b2 + -1a + b2 = 0 1a + b2 + 1-a + -b2 = 0

= 0 = 0 + 0

1a + b2 + 1-a + -b2 = 1a + -a2 + 1b + -b2

1a + b2 + 1-a + -b2 = 0.a + b

-a + -b-a + -b-1a + b2a + b

-a + -b1a + b2 + -1a + b2 = 0.1a + b2

-1a + b2

R E M A R K Notice that in the proof some of the steps involving the commutative andassociative properties of integer addition were omitted. A more detailed proof showingall the steps follows:

= 0 = a + -a = 1a + 02 + -a = 3a + 1b + -b24 + -a = 31a + b2 + -b4 + -a

1a + b2 + 1-a + -b2 = 1a + b2 + 1-b + -a2

Theorem 5–3For any integers, a and b:

1.2. -a + -b = -1a + b2

-1-a2 = a

the additive inverse of is also a. Because the additive inverse of must beunique, we have Other theorems of addition of integers can be investigated byconsidering previously developed notions. For example, we saw that andwe know that is the additive inverse of 6, or This leads us to the following:

This relationship is true in general and is stated along with its proof.

-2 + -4 = -12 + 42

2 + 4.-6

-2 + -4 = -6,

-1-a2 = a.

-a-aa + -a = 0,

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Integer SubtractionAs with integer addition, we explore several models for integer subtraction.

Chip Model for Subtraction

To find , we want to subtract (take away 2 red chips) from 3 black chips. As seenin Figure 5-9(a), if we just have 3 black chips, we can’t take 2 red ones away. Therefore, weneed to represent 3 so that at least 2 red chips are present. Recall that 1 red chip neutralizes1 black chip and so adding a black chip and a red chip (or 2 black chips and 2 red chips) isthe same as adding 0, and the problem does not change. Because we need 2 red chips, we canadd 2 black chips and 2 red chips without changing the problem. In Figure 5-9(b), we nowsee 3 represented using 5 black chips and 2 red chips. Now when the 2 red chips are “takenaway,” in Figure 5-9(c), 5 black chips are left and hence, 3 - -2 = 5.

-23 - -2

3 3 3 – –2 = 5

Figure 5-9

Charged-Field Model for Subtraction

Integer subtraction can be modeled with a charged field. For example, consider .To subtract from , we first represent so that at least 5 negative charges are pre-sent. An example is shown in Figure 5-10(a). To subtract , remove the 5 negativecharges, leaving 2 positive charges, as in Figure 5-10(b). Hence, .-3 - -5 = 2

-5

-3-3-5

-3 - -5

+ +– –– – –

(a) –3

+ +

(b) –3 – –5 = 2

Figure 5-10

(a) (b) (c)


Use Token Subtractionfor additional practiceusing the chip andcharged-field models.

Solution a.b. which can be written as or c. which can be written or 3 + x-1-32 + -1-x2,-1-3 + -x2,

-a + 4-a + -1-42,-1a + -42,3 + x

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Source: Prentice Hall Mathematics, Grade 7, 2008, Course 2 (p. 37).

School Book Page CHIP/CHARGED-FIELD MODEL

Number-Line Model for Subtraction

The number-line model used for integer addition can also be used to model integer sub-traction. While integer addition is modeled by maintaining the same direction and movingforward or backward depending on whether a positive or negative integer is added, subtrac-tion is modeled by turning around. To see how this works, examine the student page onpage 260. Study the model to make sure you are comfortable with it and use it to find5 - -3 - -2.

Notice that the chip model and the charged-field model are combined on the partial stu-dent page shown next.

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Source: Scott Foresman-Addison Wesley, Grade 5, 2008 (p. 718).

School Book Page SUBTRACTING INTEGERS

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NOW TRY THIS 5-3 Suppose a mail carrier brings you three letters, one with a check for $25 andthe other two with bills for $15 and $20, respectively. You record this as , or ; thatis, you are $10 poorer. Suppose that the next day you find out that the bill for $20 was actually intendedfor someone else and therefore you give it back to the delivery person. You record your new balance as

or as

which equals , or 10.For each of the following, make up a mail delivery story and explain how your story can help to find

the answer.

a.b. 18 - -37

23 + - 13 + - 12

25 + - 15

25 + - 15 + -20 - -20

- 10 - -20

- 1025 + - 15 + -20

Pattern Model for Subtraction

By using inductive reasoning, we can find the difference of two integers by considering thefollowing patterns, where we start with subtractions that we already know how to do. Boththe following pattern on the left and the pattern on the right start with

In the pattern on the left, the difference decreases by 1. If we continue the pattern, we haveand . In the pattern on the right, the difference increases by 1. If

we continue the pattern, we have and .

Subtraction Using the Missing-Addend Approach

Subtraction of integers, like subtraction of whole numbers, can be defined in terms ofaddition. Using the missing-addend approach, can be computed by finding a wholenumber n as follows:

Because Similarly, we compute as follows:

Because In general, for integers a and b, we have the following defi-nition of subtraction.

n = -2.5 + -2 = 3,

3 - 5 = n if, and only if, 3 = 5 + n

3 - 5n = 2.3 + 2 = 5,

5 - 3 = n if, and only if, 5 = 3 + n

5 - 3

3 - -2 = 53 - -1 = 43 - 5 = -23 - 4 = -1

3 - 5 = ? 3 - -1 = ? 3 - 4 = ? 3 - 0 = 3 3 - 3 = 0 3 - 1 = 2 3 - 2 = 1 3 - 2 = 1

3 - 2 = 1.

Definition of SubtractionFor integers a and b, is the unique integer n such that .a = b + na - bIS

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Subtraction Using Adding the Opposite Approach

Next, study the partial student page from Scott Foresman-Addison Wesley Mathematics,Grade 6, 2008 (p. 423). Consider the subtractions and additions in parts A–D and thenthe rule that the students discovered. This technique is very useful in performing integersubtractions.

R E M A R K Addition “undoes” subtraction; that is, Also, subtraction“undoes” addition; that is, 1a + b2 - b = a.

1a - b2 + b = a.

Use the definition of subtraction to compute the following:

a.b.

Solution a. Let Then so Therefore, b. Let Then so Therefore,

-2 - 10 = -12.n = -12.10 + n = -2,-2 - 10 = n.

3 - 10 = -7.n = -7.10 + n = 3,3 - 10 = n.

-2 - 103 - 10▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-5

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▲

School Book Page SUBTRACTING INTEGERS

Source: Scott Foresman-Addison Wesley, Mathematics 2008, Grade 6 (p. 423).

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The preceding theorem can be justified using the fact that the equation has aunique solution for x. From the definition of subtraction, the solution of the equation is

To show that , we only need to show that is also a solution.For that purpose, we substitute for x and check if

Consequently, a - b = a + -b.

= a = 0 + a = 1b + -b2 + a

b + 1a + -b2 = b + 1-b + a2

b + 1a + -b2 = a:a + -ba + -ba - b = a + -ba - b.

b + x = a

Theore 5–4For all integers a and b, a - b = a + -b.

NOW TRY THIS 5-4

a. Is the set of integers closed under subtraction? Why?b. Do the commutative, associative, or identity properties hold for subtraction of integers? Why?

R E M A R K Sometimes the preceding theorem is used as the definition of subtraction.

Many calculators have a change-of-sign key, either or . Other calculators use , a key that allows computation with integers. For example, to compute we

would press . Investigate what happens if you press .=3--8=+/-3-8

8 - 1-32,(-)

+/-CHS

Using the fact that compute each of the following:

a. b. c. d.

Solution a.b.c.d. -12 - 5 = -12 + -5 = -17

-12 - -5 = -12 + -1-52 = -12 + 5 = -72 - -8 = 2 + -1-82 = 2 + 8 = 102 - 8 = 2 + -8 = -6

-12 - 5-12 - -52 - -82 - 8

a - b = a + -b,

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-6

▲

▲

Use the fact that and the theorems involving the additive inverse towrite expressions equal to each of the following without parentheses.

a. b.

Solution a.b.

a + -b + - ca - 1b + c2 = a + -1b + c2 = a + 1-b + - c2 = 1a + -b2 + - c =-1b - c2 = -1b + - c2 = -b + -1- c2 = -b + c

a - 1b + c2-1b - c2

a - b = a + 1-b2

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-7

▲

▲


From our previous work with addition of integers, we know that andHence, In general, the following is true.3 - 5 = 3 + -5.3 + -5 = -2.

3 - 5 = -2

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R E M A R K It is possible to simplify the answers in Example 5-7(a) and (b) further, asfollows: and a + -b + - c = 1a - b) - c.-b + c = c + -b = c - b;

Simplify each of the following:

a. b. c.

Solution a.

= -3 + x or x - 3 = 2 + -5 + x = 2 + -5 + -1-x2

2 - 15 - x2 = 2 + -15 + -x2

-1x - y2 - y5 - 1x - 322 - 15 - x2▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼Example 5-8

▲

▲

Order of Operations

Subtraction in the set of integers is neither commutative nor associative, as illustrated inthese counterexamples:

An expression such as is ambiguous unless we know in which order to performthe subtractions. Mathematicians agree that means that is, thesubtractions in are performed in order from left to right. Similarly, means and not Thus, may be written withoutparentheses as Order of operations for integers will be revisited after multiplica-tion and division are discussed.

a - b - c.1a - b2 - c3 - 14 + 52.13 - 42 + 5

3 - 4 + 53 - 15 - 813 - 152 - 8;3 - 15 - 8

3 - 15 - 8

13 - 152 - 8 Z 3 - 115 - 82 because -20 Z -4 5 - 3 Z 3 - 5 because 2 Z -2

Compute each of the following:

a. b. c.

Solution a.b.c. 3 - 17 - 32 = 3 - 4 = -1

3 - 7 + 3 = -4 + 3 = -12 - 5 - 5 = -3 - 5 = -8

3 - 17 - 323 - 7 + 32 - 5 - 5▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-9▲

▲

TECHNOLOGY CORNER

a. On a graphing calculator or a spreadsheet, graph the function with equation b. Using the graph in (a), describe what happens as x takes on values that are less than equal to

and greater than -4.

-4,-4,y = x - -4.

b.

= 8 - x= 8 + -x= 5 + -x + 3= 5 + -x + -1-32

5 - 1x - 32 = 5 + -1x + -32

c.

= -x= -x + 0= -x + 1 y + -y2= 1-x + y2 + -y= 3-x + -1-y24 + -y

-1x - y2 - y = -1x + -y2 + -y

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Assessment 5-1A

1. Find the additive inverse of each of the following inte-gers. Write your answer in the simplest possible form.a. 2 b.c. m d. 0e. f.

2. Simplify each of the following:a. b. c.

3. Evaluate each of the following:a. b.c. d.

4. Demonstrate each of the following additions using thecharged-field or chip model:a. b.c. d.

5. Demonstrate each of the additions in problem 4 using anumber-line model.

6. Compute each of the following using a.b.c.

7. Answer each part of problem 6 using the definition ofsubtraction with the missing-addend approach.

8. Write an addition fact that corresponds to each of thefollowing sentences and then answer the question:a. A certain stock dropped 17 points and the following

day gained 10 points. What was the net change in thestock’s worth?

b. The temperature was and then it rose by What is the new temperature?

c. The plane was at 5000 ft and dropped 100 ft. What isthe new altitude of the plane?

9. On January 1, Jane’s bank balance was $300. During themonth, she wrote checks for $45, $55, $165, $35, and$100 and made deposits of $75, $25, and $400.a. If a check is represented by a negative integer and a

deposit by a positive integer, express Jane’s transactionsas a sum of positive and negative integers.

b. What was the balance in Jane’s account at the end ofthe month?

10. Use a number-line model to find the following:a. b.

11. Use patterns to show the following:a. b.

12. Perform each of the following:a.b.c.

13. In each of the following, write a subtraction problem thatcorresponds to the question and an addition problemthat corresponds to the question and then answer thequestions:

1-2 - 72 + 1038 - 1-524 - 10-2 + 13 - 102

-2 - 1 = -3-4 - -1 = -3

-4 - -3-4 - -1

8°C.-10°C

-3 - -2-3 - 23 - -2

a - b = a + -b:

-3 + -2-3 + 2-2 + 35 + -3

- ƒ 5 ƒ- ƒ -5 ƒƒ 10 ƒƒ -5 ƒ

-0-1-m)-1-2)

a + b-m

-5

a. The temperature is and is supposed to dropby midnight. What is the expected midnight

temperature?b. Moses has overdraft privileges at his bank. If he had

$200 in his checking account and he wrote a $220check, what is his balance?

14. Motor oils protect car engines over a range of tempera-tures. These oils have names like 10W–40 or 5W–30.The following graph shows the temperatures, in degreesFahrenheit, at which the engine is protected by a partic-ular oil. Using the graph, find which oils can be used forthe following temperatures:a. Between and b. Below c. Between and d. From to over e. From to 90°-8°

100°-20°50°-10°

-20°90°-5°

60°F55°F

–30° –20° –10° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100°

10W–30

10W–40

20W–30

5W–30

15. Simplify each of the following as much as possible. Showall work.a.b.

16. For which integers a, b, and c does Justify your answer.

17. Let W stand for the set of whole numbers, I the set ofintegers, the set of positive integers, and the setof negative integers. Find each of the following:a.b.c.d.e.f.

18. Let with domain I. Find the following:a. b.c. d. in terms of ae. For which values of x will the output be 3?

19. Find all integers x, if there are any, such that the follow-ing are true:a. is positive.b. is negative.c. is positive.d. ƒ x ƒ = 2.

-x - 1-x-x

f1-a2f1-22f11002f1-12

f1x2 = -x - 1I - WW - II + ¨ I -I + ´ I -W ¨ IW ´ I

I -I

+

a - 1b - c2?a - b - c =

x - 1-x - y23 - 12 - 4x2

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Assessment 5-1B

1. Find the additive inverse of each of the following inte-gers. Write your answer in the simplest possible form.a. 3 b.c. q d. 6e. f.

2. Simplify each of the following:a.b.

3. Evaluate each of the following:a. b. c.

4. Demonstrate each of the following additions using thecharged-field or chip model:a. b. c.

5. Demonstrate each of the additions in problem 4 using anumber-line model.

6. Compute each of the following using a.b.

7. Answer each part of problem 6 using the definition ofsubtraction with the missing-addend approach.

8. Write an addition fact that corresponds to each of thefollowing sentences and then answer the question:a. A visitor in a Las Vegas casino lost $200, won $100,

and then lost $50. What is the change in the gam-bler’s net worth?

5 - 1-32-3 - 5

a - b = a + -b.

-3 + -3-5 + 2-2 + 5

- ƒ -3 ƒƒ 15 ƒƒ -3 ƒ

-1-x2-1-52

3 + x-n

-4

b. In four downs, the football team lost 2 yd, gained7 yd, gained 0 yd, and lost 8 yd. What is the total gainor loss?

9. Use a number-line model to find the following:a. b.

10. Use patterns to show the following:a. b.

11. Perform each of the following:a.b.c.

12. Answer each of the following:a. In a game of Triominoes, Jack’s scores in five succes-

sive turns are and 45. What is his totalat the end of five turns?

b. The largest bubble chamber in the world is 15 ft indiameter and contains 7259 gal of liquid hydrogen ata temperature of If the temperature isdropped by per hour for 2 consecutive hours,what is the new temperature?

c. The greatest recorded temperature ranges in theworld are around the “cold pole” in Siberia. Temper-atures in Verkhoyansk have varied from to

What is the difference between the high andlow temperatures in Verkhoyansk?98°F.

-94°F

11°C-247°C.

17, -8, -9, 14,

-2 - 7 + 38 - 11 - 10-2 - 17 + 102

-3 - 2 = -5-2 - -3 = 1

-4 - 3-3 - -2

20. Let with domain I. Find the following:a. b.c. All the inputs for which the output is 1d. The range

21. In each of the following, find all integers x satisfying thegiven equation:a.b.c.

22. Determine how many integers there are between the fol-lowing given integers (not including the given integers):a. 10 and 100 b. and

23. Suppose and Insert paren-theses in the expression to obtain thegreatest possible and the least possible values. What arethese values?

24. An arithmetic sequence may have a positive or negativedifference. In each of the following arithmetic sequences,find the difference and write the next two terms:a.b. x + y, x, x - y

0, -3, -6, -9

a - b - c - dd = -3.a = 6, b = 5, c = 4,-10-30

ƒ -x ƒ = ƒ x ƒƒ x ƒ + 2 = 10ƒ x - 6 ƒ = 6

f 1-12f 1102f (x2 = ƒ 1 - x ƒ 25. In an arithmetic sequence, the eighth term minus the

first term equals 21. The sum of the first and the eighthterm is Find the fifth term of the sequence.

26. Classify each of the following as true or false. If false,show a counterexample that makes it false.a.b.c.

27. Solve the following equations:a. b.c.

28. Complete each of the following integer arithmetic prob-lems on your calculator, making use of the change-of-signkey. For example, on some calculators to find press .a. b.c. d.e. 16 - -7

-12 - 627 + -5-12 + 6-12 + -6

=+/-4++/-5

-5 + -4,

-x = 5-10 + x = -7x + 7 = 3

ƒ -x + -y ƒ = ƒ x + y ƒƒ x - y ƒ = ƒ y - x ƒƒ -x ƒ = ƒ x ƒ

-5.

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13. Simplify each of the following as much as possible. Showall work.a. b.

14. Let W stand for the set of whole numbers, I the set of in-tegers, the set of positive integers, and the set ofnegative integers. Find each of the following:a. b. c.

15. a. Prove that for all integers x and b. Does part (a) imply that subtraction is commutative?

Explain.16. Complete the magic square using the following integers:

-13, -10, -7, -4, 2, 5, 8, 11.

y.-x - y = -y - x;I ¨ IW - I -W - I +

I -I

+

4x - 12 - 3x24x - 2 - 3x

22. Determine how many integers there are between the fol-lowing given integers (not including the given integers):a. and 10b. x and y (if )

23. From midnight to 1:00 A.M in January, the temperaturedropped . After it dropped, the outside temperaturewas . What was the temperature at midnight?

24. An arithmetic sequence may have a positive or negativedifference. In each of the following arithmetic sequences,find the difference and write the next two terms:a.b.

25. Find the sums of the following arithmetic sequences:a.b.c.

26. Classify each of the following as true or false. If false,show a counterexample that makes it false.a. b.c.

27. Solve the following equations:a. b.c.

28. Assume that gear A has 56 teeth and gear B has 14 teeth.Suppose that the number of counterclockwise rotationsis designated by a positive number and the number ofclockwise rotations by a negative number. If gear Arotates 7 times per minute, how many times per minutedoes gear B rotate? Explain your reasoning.

- x - 8 = -

91 - x = -13-

x + 5 = 7

ƒ x3 ƒ = x2 ƒ x ƒƒ x3 ƒ = x3ƒ x2 ƒ = x2

100 + 98 + 96 + Á + -6100 + 99 + 98 + Á + -50-20 + -19 + -18 + Á + 18 + 19 + 20

1 - 3x, 1 - x, 1 + x7, 3, -1, -5

-12°F15°F

x 6 y-10

–1

17. Let with domain I. Find the following:a. b.c. d. in terms of ae. For which values of x will the output be

18. Find all integers x, if there are any, such that the follow-ing are true:a.b. is negative.c. is positive.d. is positive.e. is negative.

19. Let with domain I. Find the following:a. b.c. All the inputs for which the output is 7d. The range

20. a. For each of the following functions, find i.

ii.iii.

b. Interpret your answers in part (a) using the functionmachine model.

c. Find other functions for which Justifyyour answer.

21. By the definition of absolute value, the function can be written as follows:

Write the function in a similar waywithout absolute value.

f1x2 = ƒ x - 6 ƒ

f1x2 = ex, if x Ú 0

-x, if x 6 0

ƒ x ƒf1x2 =

f 1 f 1x22 = x.

f 1x2 = -x + 2f 1x2 = -xf 1x2 = x

f 1 f1x22:

f 1-12f 1102f 1x2 = ƒ x - 5 ƒ

-x - 1-x - 1- ƒ x ƒ- ƒ x ƒ- ƒ x ƒ = 2.

-11?f 1-a2f 1-22f 11002f 1-12

f 1x2 = -3x - 2

A B

29. Estimate each of the following and then use a calculatorto find the actual answer:a.b.c.d.

30. Find a quick way to calculate the following:

2006 + 2007?1 - 2 + 3 - 4 + 5 - 6 + Á - 2004 + 2005 -

-301 - -1303 + 4993992 - -10,003 - 101-1992 + 3005 - 497343 + -42 - 402

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Mathematical Connections 5-1

Communication1. A turnpike driver had car trouble. He knew that he had

driven 12 mi from milepost 68 before the trouble started.Assuming he is confused and disoriented when he calls onhis cellular phone for help, how can he determine his pos-sible location? Explain.

2. Dolores claims that the best way to understand thatfor all integers a and b, is to show

that when you add b to each expression you get equalanswers.a. Explain why you think Dolores is making this claim.b. Do you agree with Dolores that her approach is the

“best way”? If not, what is a better approach?3. Addition of integers with like signs can be described using

absolute values as follows:

To add integers with like signs, add the absolute values ofthe integers. The sum has the same sign as the integers.

Describe in a similar way how to add integers with un-like signs.

4. Explain why and are additive inverses ofeach other.

5. a. The absolute value of an integer is never negative.Does this contradict the fact that the absolute value ofx could be equal to Explain why or why not.

b. Explain how to write the additive inverse ofusing the least number of symbols.

6. If an integer a is pictured on the number line, then the dis-tance from the point on the number line that representsthe integer to the origin is Using this idea, answer thefollowing:a. Explain why is the distance between the

points that represent the integers a and b.b. One way to define “less than” for integers is as follows:

if, and only if, a is to the left of b on the numberline. Consequently, if, and only if, b is to theright of a. Use these ideas to mark on a number line allintegers x such thati. ii.

iii. iv.7. Recall the definition of less than for whole numbers using

addition and define when a and b are any integers.Use your definition to show that

Open-Ended8. Describe a realistic word problem that models 1-852 - 1-302.

-50 +

-8 6 -7.a 6 b

ƒ x ƒ 7 - 1.ƒ x ƒ Ú 5.

ƒ x ƒ 6 1.ƒ x ƒ 6 5.

b 7 aa 6 b

ƒ a - b ƒ

ƒ a ƒ .

a - b - c

-x?

a - bb - a

a - b = a + -b,

9. In a library some floors are below ground level and oth-ers are above ground level. If the ground-level floor isdesignated the zero floor, design a system to number thefloors.

10. a. I am choosing an integer. I then subtract 10 from theinteger, take the opposite of the result, add andfind the opposite of the new result. My result is What is the original number?

b. Judy wants to do the activity in part (a) with her class-mates. Each classmate probably chooses a differentnumber and Judy wants to tell each classmate quicklywhat number was chosen. Judy figures out that theonly thing she needs to do is to add 7 to each answershe gets. Does this always work? Explain why or whynot.

c. Come up with your own “trick” similar to the one inpart (b) that works for each answer you get from yourclassmates.

11. a. Write a function such that for all integer inputs the outputs will be negative.

b. Write a function such that the sequenceis an arithmetic sequence.

Cooperative Learning12. Examine several elementary mathematics textbooks. Re-

port on how addition and subtraction of integers istreated, and on how various properties are justified. Dis-cuss in your group how the treatment of addition andsubtraction of integers presented in this section comparesto the treatment in elementary textbooks.

13. Look at several history of mathematics books and theInternet and report in your group on when and hownegative integers were introduced first.

14. Number each card in a set of 21 3-by-5 cards with an in-teger from to 10. Lay the cards on the floor to forma number line. Choose someone from your group to actlike the hiker on the student pages for number-lineaddition and subtraction. Give the hiker directionsto walk the number line to solve problems 5 and 9 inProblem Set 5-lA. Try other addition and subtractionproblems to make sure that the number-line model isunderstood by everyone in your group and could be usedin an elementary classroom.

-10

f1-12, f1-22, f1-32, Áf1x2

f1x2

-3.-3,

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Section 5-2 Multiplication and Division of Integers 269

Questions from the Classroom15. A fourth-grade student devised the following subtrac-

tion algorithm for subtracting 4 minus 7 equals negative 3.

80 minus 20 equals 60.

60 plus negative 3 equals 57.

Thus the answer is 57. What is your response as ateacher?

16. An eighth-grade student claims she can prove that subtrac-tion of integers is commutative. She points out that if aand b are integers, then Since addition iscommutative, so is subtraction. What is your response?

17. A student had the following picture of an integer and itsopposite. Other students in the class objected, saying that

should be to the left of 0. How do you respond?-a

a - b = a + -b.

84- 27

-3+ 60

57

84- 27

-360

84- 27

-3

84 - 27:

a 0 –a

18. A student found that addition of integers can be per-formed by finding the sum or the difference of theabsolute values of these integers and then attaching the“ ” sign if necessary. She would like to know if this isalways true. How do you respond?

-

10

�10

0

10

�10

0

10

�10

0

10

�10

0

At recess, the temperature was 5 degrees.

How many degrees did the temperature rise?a. 2 degrees b. 3 degrees c. 5 degrees d. 8 degreesTIMSS, Grade 4, 2003

National Assessment of Educational Progress (NAEP) Question

Paco had 32 trading cards. He gave N trading cards tohis friend. Which expression tells how many tradingcards Paco has now?a. b. c. d.NAEP, Grade 4, 2007

32 , NN - 3232 - N32 + N

BRAIN TEASER If the digits 1 through 9 are written in order, it is possible to place plus and minus signsbetween the numbers or to use no operation symbol at all to obtain a total of 100. For example,

Can you obtain a total of 100 using fewer plus or minus signs than in the given example? Note that digits,such as 7 and 8 in the example, may be combined.

1 + 2 + 3 - 4 + 5 + 6 + 78 + 9 = 100

Third International Mathematics and Science Study(TIMSS) Question

When Tracy left for school, the temperature was minus3 degrees.

Multiplication and Division of Integers

We approach multiplication of integers through a variety of models: patterns, charged-field,chip, and number-line. Note that the reasoning with these models is inductive reasoning andtherefore does not constitute a proof.

5-2

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Patterns Model for Multiplication of Integers

We may approach multiplication of integers by using repeated addition. For example, if arunning back lost 2 yd on each of three carries in a football game, then he had a net loss of

or yards. Since can be written as using re-peated addition, we have

Consider It is meaningless to say that there are threes in a sum. But if thecommutative property of multiplication is to hold for all integers, we must have

Next, consider We can develop the following pattern:

The first four products, and 0, are terms in an arithmetic sequence with a fixeddifference of 2. If the pattern continues, the next three terms in the sequence are 2, 4, and6. Thus it appears that Likewise, 1-221-32 = 6.1-321-22 = 6.

-6, -4, -2,

-31-22 = ? -21-22 = ? -11-22 = ?

01-22 = 0 11-22 = -2 21-22 = -4 31-22 = -6

1-321-22.1-223 = 31-22 = -6.

-21-223.31-22 = -6.

31-22,-2 + -2 + -2-6,-2 + -2 + -2,

R E M A R K Notice the phrase “it appears that ” Later in this section, weexplore why -31-22 = 6.

1-321-22 = 6.

Next we approach multiplication of integers using the chip model, the charged-fieldmodel, and the number-line model. In all of these models we start with 0, represented pos-sibly in various ways.

Chip Model and the Charged-Field Model for Multiplication

The chip model and the charged-field model can both be used to illustrate multiplication of in-tegers. Consider Figure 5-11(a), where is pictured using a chip model. The product

is interpreted as putting in 3 groups of 2 red chips each. In Figure 5-11(b), ispictured as 3 groups of 2 negative charges.

31-2231-2231-22

3 groups of2 red chips

3(–2) = –6

– –– –– –

3 groups of 2negative charges

3(–2) = –6

Figure 5-11


TECHNOLOGY CORNER On a spreadsheet, in column A enter 5 as the first entry and then write aformula to add to 5 for the second entry. Then add to the second entry and fill down, continuingthe pattern. In column B, repeat the process. In column C, find the product of the respective entries incolumns A and B. What patterns do you observe?

- 1- 1

(a) (b)

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To find using the chip model, we interpret the signs as follows: is taken tomean “remove 3 groups of ”; is taken to mean “2 red chips.” To do this, we start with avalue of 0 that includes at least 6 red chips, as shown in Figure 5-12(a). When we remove 6red chips, we are left with 6 black chips. The result is a positive 6, so Simi-lar reasoning can be used in Figure 5-12(b) with the charged-field model.

1-321-22 = 6.

-2

-31-321-22


Remove 3groups of

2 red chips.

60

Remove 3 groupsof 2 negative

charges.

–––

– ––

+0 Charge 6 Charge

+

+ + + +

+ +

+ + + +

+

Figure 5-12

Number-Line Model

As with addition and subtraction, we demonstrate multiplication by using a hiker movingalong a number line, according to the following rules:

1. Traveling to the left (west) means moving in the negative direction, and traveling to theright (east) means moving in the positive direction.

2. Time in the future is denoted by a positive value, and time in the past is denoted by anegative value.

Consider the number line shown in Figure 5-13. Various cases using this number line aregiven next.

1. If you are now at0 and move east at3 mph, where willyou be 4 hr from now?

2. If you are now at0, moving east at3 mph, wherewere you 4 hr ago?

3. If you are now at0 and move west at3 mph, wherewill you be 4 hrfrom now?

4. If you are now at0, moving west at3 mph, wherewere you 4 hr ago?

Move eastat 3 mph.

×

4 hrfrom now

=

You will be12 mi east of 0.

3 4 12

× =

× =3 4 12– –

Moving eastat 3 mph.

×

4 hrago

=

You were 12 miwest of 0.

3 4 12– –

Move westat 3 mph.

4 hrfrom now

You will be12 mi west of 0.

3– 124–

Moving westat 3 mph.

4 hrago

You were 12 mieast of 0.

–18 –15 –12 –9 –6 –3 0 3 6 9 12 15 18W E

Figure 5-13

(a) (b)

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An alternative use of the number line to show multiplication of integers is given inFigure 5-14.

–5 0 1 2 3 4 5 6–4 –3 –2

+2 +2 +2

–1–6

–5 0 1 2 3 4 5 6–4 –3 –2 –1–6

–2 –2 –2

–5 0 1 2 3 4 5 6–4 –3 –2 –1–6

+2 +2 +2

–5 0 1 2 3 4 5 6–4 –3 –2 –1–6

–2 –2 –2

Figure 5-14

a. means three groups of 2 each: 3 # 2 = 6.3 # 2

b. means three groups of each: 31-22 = -6.-231-22

c. The integers 3 and are opposites. You can think of as theopposite of three groups of 2 each. So 1-322 = -6.

1-322-3

d. You can think of as the opposite of three groups of each. Since 31-22 = -6, -31-22 = 6.

-21-321-22

These models illustrate the following:

Theorem 5–5For any whole numbers a and b, the following hold:

1.2. 1-a2b = b1-a2 = -1ab21-a21-b2 = ab

R E M A R K We will show later in this section that this theorem is true for all integers aand b.

Properties of Integer MultiplicationThe set of integers has properties under multiplication analogous to those of the set ofwhole numbers under multiplication. These properties are summarized next.

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An approach to showing that uses the uniqueness theorem of additiveinverses. If we can show that and are additive inverses of the same number,then they must be equal. By definition, the additive inverse of is That

is also the additive inverse of can be proved by showing that The proof follows:

Distributive property of multiplication over additionAdditive inverseZero multiplication

Now we have

Because and are both additive inverses of and because the additive in-verse must be unique, Using this approach, we could prove the follow-ing theorem (the proof is explored in Assessment 5-2A):

1-223 = -12 # 32.12 # 32-12 # 321-223

-12 # 32 + 2 # 3 = 0 1-223 + 2 # 3 = 0

= 0 = 0 # 3

1-223 + 2 # 3 = 1-2 + 223

1-223 + 2 # 3 = 0.2 # 31-223

-12 # 32.12 # 32-12 # 321-223

1-223 = -12 # 32


Theorem 5–7For every integer a, 1-12a = -a.

It is important to keep in mind that is true for all integers a. Thus if wesubstitute for a, we get Because we have anotherjustification for the fact that Using this result, the preceding property, and1-121-12 = 1.

-1-12 = 1,1-121-12 = -1-12.-11-12a = -a

Theorem 5–6: Properties of Integer MultiplicationThe set of integers I satisfies the following properties of multiplication for all integers

Closure property of multiplication of integers ab is a unique integer.Commutative property of multiplication of integersAssociative property of multiplication of integersMultiplicative identity property 1 is the unique integer such that for all integers

Distributive properties of multiplication over addition for integersand Zero multiplication property of integers 0 is the unique integer such that for all integers a # 0 = 0 = 0 # a.

a,1b + c2a = ba + ca.

a1b + c2 = ab + ac1 # a = a = a # 1.

a,1ab2c = a1bc2.

ab = ba.

a, b, c � I:

HistoricalNote

Emmy Noether (1882–1935) made lasting contributions to the study of rings, algebraicsystems among which is the set of integers. When she entered the University of Erlangen(Germany) in 1900, Emmy Noether was one of only two women enrolled. Aftercompleting her doctorate in 1907, she could not find a suitable job because she was awoman. In 1919, she got a university appointment without pay and only later a very mod-est salary. In 1933, along with many other scholars, she was dismissed from the Univer-sity at Göttingen because she was Jewish. She emigrated to the United States and taughtat Bryn Mawr College until her untimely death only 18 months after arriving in theUnited States.

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the properties of integers listed earlier, we can show that and thatfor all integers a and b as follows:

Also:

We have established the following theorems:

= ab = 11ab2 = 31-121-1241ab2

1-a21-b2 = 31-12a431-12b4

= -1ab2 = 1-121ab2

1-a2b = 31-12a4b

1-a21-b2 = ab1-a2b = -1ab2

Theorem 5–8For all integers a and b,

1-a21-b2 = ab 1-a2b = -1ab2

R E M A R K It is important to note that in Theorem 5–8, and are not necessarilynegative and a and b are not necessarily positive.

-b-a

The distributive property of multiplication over subtraction follows from the distributiveproperty of multiplication over addition:

Consequently, From this and the commutative property of multiplica-tion we see that 1b - c2a = ba - ca.

a1b - c2 = ab - ac.

= ab - ac = ab + -1ac2 = ab + a1- c2

a1b - c2 = a1b + - c2

Simplify each of the following so that there are no parentheses in the final answer:

a. b.

Solution a.b.

(Note: means )

Thus, 1a + b21a - b2 = a2 - b2.

= a2 - b2 = a2 + 0 + -b2

-1b22.-b2 = a2 + ab + -1ab2 + -b2 = a2 + ba + -1ab + b22

= 1a2 + ba2 - 1ab + b22

1a + b21a - b2 = 1a + b2a - 1a + b2b1-321x - 22 = 1-32x - 1-32122 = -3x - 1-62 = -3x + -1-62 = -3x + 6

1a + b21a - b21-321x - 22▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-10

▲

▲

Theorem 5–9: Distributive Property of Multiplication over Subtraction for Integers

For any integers a, b, and c,a1b - c2 = ab - ac and 1b - c2a = ba - ca

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The result in Example 5-10(b) is the difference-of-squaresformula.

1a + b21a - b2 = a2 - b2


Use the difference-of-squares formula to simplify the following:

a. b. c.

Solution a.b.c.

= 12x= 2x # 6= 2x1x + 3 - x + 32

1x + 322 - 1x - 322 = 31x + 32 + 1x - 324 31x + 32 - 1x - 3241-4 + b21-4 - b2 = 1-422 - b2 = 16 - b214 + b214 - b2 = 42 - b2 = 16 - b2

1x + 322 - 1x - 3221-4 + b21-4 - b214 + b214 - b2▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-11

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▲

NOW TRY THIS 5-5 Determine how to use the difference-of-squares formula to compute thefollowing mentally:

a. b. c. d. 998 # 100224 # 3622 # 18101 # 99

When the distributive property of multiplication over subtraction is written in reverseorder as

and similarly for addition, the expressions on the right of each equation are in factored form.We say that the common factor a has been factored out. Both the difference-of-squares for-mula and the distributive properties of multiplication over addition and subtraction can beused for factoring.

ab - ac = a1b - c2 and ba - ca = 1b - c2a

Factor each of the following completely:

a. b. c. d. e.

Solution a.b.c.d.e. 5x2 - 2x2 = 15 - 22x2 = 3x2

3x - 6 = 31x - 22

-3x + 5xy = x1-3 + 5y21x + y22 - z2 = 1x + y + z21x + y - z2x2 - 9 = x2 - 32 = 1x + 321x - 32

5x2 - 2x23x - 6-3x + 5xy1x + y22 - z2x2 - 9▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-12

▲

▲

Integer DivisionIn the set of whole numbers, where is the unique whole number c such that

If such a whole number c does not exist, then is undefined. Division on theset of integers is defined analogously.

a , ba = bc.b Z 0,a , b,

Definition of Integer DivisionIf a and b are any integers, then is the unique integer c, if it exists, such that a = bc.a , b

R E M A R K Notice that if it exists, is the solution of a = bx.a , b,ISB

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R E M A R K Notice that from Example 5-14(e) and (f), we have Byconvention, means and means and thereforeequals -1x # x # x # x2.

-1x42-x41-x21-x21-x21-x21-x241-324 Z -34.

Example 5-13 suggests that the quotient of two negative integers, if it exists, is a positive inte-ger and the quotient of a positive and a negative integer, if it exists, or of a negative and a positiveinteger, if it exists, is negative.

Use the definition of integer division, if possible, to evaluate each of the following:

a. b. c. d.e. f.

Solution a. Let Then and consequently Thus,

b. Let Then and therefore Thus,

c. Let Then and consequently Thus,

d. Let Then Because no integer c exists to satisfy thisequation (why?), we say that is undefined over the set of integers.

e. Let Then and consequently f. Let Then and hence x = b.ab = ax1ab2 , a = x.

x = a.ab = bx1ab2 , b = x.

-12 , 5

-12 = 5c.-12 , 5 = c.

-12 , 1-42 = 3.c = 3.-12 = -4c-12 , 1-42 = c.

-12 , 4 = -3.c = -3.-12 = 4c-12 , 4 = c.

12 , 1-42 = -3.c = -3.12 = -4c12 , 1-42 = c.

1ab2 , a, a Z 01ab2 , b, b Z 0

-12 , 5-12 , 1-42-12 , 412 , 1-42▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-13

▲

▲

NOW TRY THIS 5-6 Use the definition of division for integers to show that dividing by 0 is notpossible.

Order of Operations on IntegersWhen addition, subtraction, multiplication, division, and exponentiation appear withoutparentheses, exponentiation is done first in order from right to left, then multiplicationsand divisions in the order of their appearance from left to right, and then additions andsubtractions in the order of their appearance from left to right. Arithmetic operations thatappear inside parentheses must be done first.

Evaluate each of the following:

a. b. c.d. e. f.

Solution a.b.c.d.e.f. -34 = -1342 = -1812 = -811-324 = 1-321-321-321-32 = 812 + 16 , 4 # 2 + 8 = 2 + 4 # 2 + 8 = 2 + 8 + 8 = 10 + 8 = 182 - 3 # 4 + 5 # 2 - 1 + 5 = 2 - 12 + 10 - 1 + 5 = 412 - 524 + 1 = 1-324 + 1 = -12 + 1 = -112 - 5 # 4 + 1 = 2 - 20 + 1 = -18 + 1 = -17

-341-3242 + 16 , 4 # 2 + 82 - 3 # 4 + 5 # 2 - 1 + 512 - 524 + 12 - 5 # 4 + 1▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-14▲

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Ordering IntegersAs with whole numbers, a number line as shown in Figure 5-15 can be used to describegreater-than and less-than relations for the set of integers. Because is to the left of on the number line, we say that “ is less than ” and we write We can alsosay that “ is greater than ” and we can write -3 7 -5.-5,-3

-5 6 -3.-3,-5

-3-5


–5 0 1 2 3 4 5–4 –3 –2 –1

Figure 5-15

Notice that since is to the left of there is a positive integer that can be added toto get namely, 2. Thus, because The definition of less

than for integers is similar to that for whole numbers.

-5 + 2 = -3.-3-5 6-3,-5

-3,-5

Definition of Less Than for IntegersFor any integers a and b, a is less than b, written if, and only if, there exists a positiveinteger k such that a + k = b.

a 6 b,

The last equation implies that Thus we have proved the following theorem:k = b - a.

Theorem 5–10(or equivalently, ) if, and only if, is equal to a positive integer; that is, is

greater than 0.b - ab - ab 7 aa 6 b

Using this theorem, because (Also means that Notice that if, and only if, Also if,and only if, )

The preceding theorem can be used to justify each of the following for integers x, y, and n:a … b.

b Ú aa 6 b.b 7 aa 6 b or a = b.a … b

-3 - 1-52 = -3 + -1-52 = -3 + 5 = 2 7 0.-3-5 6

Theorem 5–11a. If and n is any integer, then b. If then c. If and then d. If and then nx 7 ny.n 6 0,x 6 y

nx 6 ny.n 7 0,x 6 y-x 7 -y.x 6 y,

x + n 6 y + n.x 6 y

The justifications are given next.

a. Because We need to show that We haveBecause we have

and hence b. Because We need to show that We have

Because wehave and hence -x 7 -y.-x - 1-y2 7 0,

y - x 7 0,-x + -1-y2 = -x + y = y + -x = y - x.-x - 1-y2 =-x - 1-y2 7 0.y - x 7 0.x 6 y,

x + n 6 y + n.1x + n2 7 0,y + n -y - x 7 0,y + n - 1x + n2 = y + n - x - n = y - x.

1y + n2 - 1x + n2 7 0.x 6 y, y - x 7 0.

Research NoteStudents do not havea good understandingof the concepts ofequivalent equations.For example, thoughable to usetransformations tosolve simple equations

becomes,

students seemunaware that eachtransformationproduces anequivalent equation(Steinberg et al.1990).

x + 2 - 2 = 5 - 221x + 2 = 5

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c. Because We need to show that We have Because n is a positive integer and is positive, must also be

positive. Because we have d. To show that we need only show that We have

Since (why?). Because and is positive. Thus, and hence,

From the Research Note on page 277 we see that students do not have good understandingof the concept of equivalent equations. Practice on this concept is given in Example 5-15.

nx 7 ny.nx - ny 7 0x - y 6 0, n1x - y2n 6 0y - x 7 0, x - y 6 0n1x - y2.

nx - ny =nx - ny 7 0.nx 7 ny,nx 6 ny.ny - nx 7 0,

n1y - x2y - xn1y - x2.ny - nx =ny - nx 7 0.x 6 y, y - x 7 0.

Use the theorems developed above to find all integers x that satisfy the following:

a.b.c. If find the values of

Solution a. If then by Theorem 5–11 (a),

We can also write the solution set (the set of all solutions) as

Notice that, strictly speaking, we have only shown that every x that satisfiesthe first inequality also satisfies To be sure that representsall the solutions, we need to show the converse; that is, if then

This can be easily done by adding 3 to both sides of b. If then

c. If then

5 - 3x Ú 11; that is, all integers in the set 511, 12, 13, 14, Á 6. 5 - 3x Ú 5 + 6

5 + -3x Ú 5 + 6 -3x Ú 6 -3x Ú -31-22

x … -2,

x 7 -8, x is an integer -1-x2 7 -8 by Theorem 5–11 1b2

- x 6 8 -x - 3 + 3 6 5 + 3

-x - 3 6 5,x 6 -5.x + 3 6 -2.

x 6 -5x 6 -5x 6 -5.

5-6,-7,-8,-9, Á 6

x 6 -5, x is an integer. x + 3 + -3 6 -2 + -3

x + 3 6 -2

5 - 3x.x … -2,

-x - 3 6 5x + 3 6 -2▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-15

▲

▲

Extending the Coordinate System

We extended a number line to include all integers. This new extended number line canbecome the x- and y-axes of a coordinate system. In Chapter 14 we investigate the coordi-nate system for all real numbers. Meanwhile, look at the student page where the coordinatesystem is extended to include negative integers. Answer the questions on the student page.Notice that when the descriptions left or right of the axis and similarly for are not true.

y = 0y-x = 0,

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School Book Page THE COORDINATE SYSTEM

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NOW TRY THIS 5-7 In each of the following find and graph all points , satisfying the given con-dition, where x and y are integers.

a. b. c. d. ƒ x ƒ + ƒ y ƒ = 5y = ƒ x ƒ y = -xy = x

y21x

BRAIN TEASER Express each of the numbers from 1 through 10 using four 4s and any operations. For example,

1 = -4 + 4 + 14 , 42 1 = 14 , 4244, or 1 = 44 , 44, or

Assessment 5-2A

1. Use patterns to show that 2. Use the charged-field model to show that 3. Use the number-line model to show that 4. In each of the following charged-field models, the encir-

cled charges are removed. Write the corresponding in-teger multiplication problem with its solution based onthe model.a. b.

21-42 = -8.1-421-22 = 8.

1-121-12 = 1. d.e.

8. Evaluate each of the following products and then, if pos-sible, write two division statements that are equivalent tothe given multiplication statements. If two divisionstatements are not possible, explain why.a.b.c.d.

9. In each of the following, x and y are integers Usethe definition of division in terms of multiplication toperform the indicated operations. Write your answers insimplest form.a.b.

10. In a lab, the temperature of various chemical reactions ischanging by a fixed number of degrees per minute.Write a numeric expression that describes each of thefollowing:a. The temperature at 8:00 P.M. is If it dropped

per minute, what is the temperature at 8:30 P.M.?b. The temperature at 8:20 P.M. is If it dropped

per minute, what was the temperature at 7:55 P.M.?c. The temperature at 8:00 P.M. is If it is dropping

per minute, what was the temperature at 7:30 P.M.?d. The temperature at 8:00 P.M. is If it increased

every minute by what was the temperature at 7:40 P.M.?

11. If it was predicted that the farmland acreage lost tofamily dwellings over the next 9 years would be 12,000acres per year, how much acreage would be lost tohomes during this time period?

3°C,25°C.

4°C-20°C.

4°C0°C.

3°C32°C.

1-xy2 , y14x2 , 4

y Z 0.0 # 01-3201-521-421-625

ƒ -24 ƒ , 3419 - 15241-6 + 62 , 1-2 + 22

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5. The number of students eating in the school cafeteriahas been decreasing at the rate of 20 per year for manyyears. Assuming this trend continues, write a multiplica-tion problem that describes the change in the number ofstudents eating in the school cafeteria for each of thefollowing:a. The change over the next 4 yearsb. The situation 4 years agoc. The change over the next n yearsd. The situation n years ago

6. Use the definition of division to find each quotient, ifpossible. If a quotient is not defined, explain why.a. b.c.

7. Evaluate each of the following, if possible:a.b.c. -8 , 1-8 + 821-10 # 52 , 51-10 , -221-22

-5 , 0-143 , 13-40 , -8 IS

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12. Show that the distributive property of multiplicationover addition, is true for each of thefollowing values of a, b, and c:a.b.

13. Compute each of the following:a. b.c.d.e. f.g. h.

14. Compute the following without using a calculator:a. b.c.

15. If x is an integer and which of the following arealways positive and which are always negative?a. b. c.d. e.

16. Which of the expressions in problem 15 are equal toeach other for all values of x?

17. Identify the property of integers being illustrated in eachof the following:a.b.c.d.

18. Simplify each of the following:a. b.c. d.

19. Multiply each of the following and combine terms wherepossible:a. b.c.d.

20. Find all integers x (if possible) that make each of thefollowing true:a. b.c. d.e. f.g.h.

21. Solve each of the following for x.a.b.c.d.e. f.g.h.

22. Use the difference-of-squares formula to simplify eachof the following, if possible:a.b.c. 1-x - y21-x + y215 - 100215 + 100252 # 48

1x - 121x + 32 = 01x - 122 = 1x + 322

1x - 122 = 9x2 = 4

-215x - 62 - 30 = -x3x - x - 2x = 3-215x - 32 = 26-3x - 8 = 7

0 , x = 0x , 1-x2 = -1

x , 1-32 = -2x , 3 = -125x = -30-2x = 0-3x = -6-3x = 6

-21x + y - z2-x1x - y2

x1x - y2-21x - y2

-1 # x-21-x + y2 + x + y-2x1-y21-x21-y2

1-9235 + 1-824 = 1-92 # 5 + 1-921-825341-324 = 15 # 421-321-421-72 � I1-3214 + 52 = 14 + 521-32

1-x23-x31-x22x2-x2

x Z 0,-1-225 + 0 # 9 - ƒ 7 - 15 ƒ - 15

-28 + 281-2264 - 264

10 - 3 # 7 - 41-22 + 3-2 + 3 # 5 - 11-121511-1250

1-325 , 1-321-1025 , 1-1022

1-2241-223

a = -3, b = -3, c = 2a = -1, b = -5, c = -2

a1b + c2 = ab + ac,23. Factor each of the following expressions completely.

a.b.c.d.e.f.g.

24. a. Use the distributive property of multiplication overaddition or over subtraction and other properties toshow that

b. Use your results from (a) to compute each of the fol-lowing mentally:(i) (Hint: Write )(ii)

(iii)25. In each of the following, find the next two terms. As-

sume the sequence is arithmetic or geometric, and findits difference or ratio and the nth term.a.b.c.

26. Find the sum of the first 100 terms in the arithmeticsequence

27. Find the first five terms of the sequences whose nthterm isa.b.c.

28. Find the first two terms of an arithmetic sequence inwhich the fourth term is and the 101st term is

29. Tira noticed that every 30 sec, the temperature of achemical reaction in her lab was decreasing by the samenumber of degrees. Initially, the temperature was and 5 min later, In a second experiment, Tiranoticed that the temperature of the chemical reactionwas initially and was decreasing by everyminute. If she started the two experiments at the sametime, when were the temperatures of the reactions thesame? What was that temperature?

30. Find all integer values (if any) of and for which thefollowing are true.a.b.c. x2 7 y2.

-x2 = x2xy = - ƒ x ƒ ƒ y ƒ

yx

3°C-57°C

-12°C.28°C

-493.-8

1-22n - 1.-5n + 3.n2 - 10.

-4, -1, 2, 5, Á .-10, -7,

2, -22, 23, -24, 25, -26, _, _-2, -4, -8, -16, -32, -64, _, _-10, -7, -4, -1, 2, 5, _, _

9972992

98 = 100 - 2.982

1a - b22 = a2 - 2ab + b2

4x2 - 25y216 - a2abc + ab - a3xy + 2x - xzx2 + xyxy + x3x + 5x

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Assessment 5-2B

1. Use patterns to show that 2. Use the charged-field model to show that 3. Use the number-line model to show that 4. In each of the following charged-field models, the encir-

cled charges are removed. Write the corresponding in-teger multiplication problem with its solution based onthe model.a. b.

21-22 = -4.1-221-22 = 4.

1-221-22 = 4. c. The temperature at 8:00 P.M. is If it increasedevery minute by d degrees, what was the temperaturem minutes before?

11. a. On each of four consecutive plays in a football game,a team lost 11 yd. If lost yardage is interpreted as anegative integer, write the information as a productof integers and determine the total number of yardslost.

b. If Jack Jones lost a total of 66 yd in 11 plays, howmany yards, on the average, did he lose on eachplay?

12. Show that the distributive property of multiplicationover addition, is true for each of thefollowing values of a, b, and c:a.b.

13. Compute each of the following:a. b.c. d.e. f.g. h.

14. Compute the following without using a calculator:a.b.

15. If x is an integer and which of the following arealways positive and which are always negative?a. b. c.d. x e.

16. Which of the expressions in problem 15 are equal toeach other for all values of x?

17. Identify the property of integers being illustrated in eachof the following:a.b.c.d.

18. Simplify each of the following:a.b.c.d.

19. Multiply each of the following and combine terms wherepossible:a.b.c.d.

20. Find all integers x (if possible) that make each of the fol-lowing true:a. x , 0 = 1

1-x2 + 221x2 - 121x - y - 121x + y + 121-5 - x215 + x2-x1x - y - 32

-11x - y2 + xy - 1y - x21-22a - 1a - b21-12x - 21-y2

1-223 = 31-22-213 + 42 = -2132 + 1-2241-420 = 01-22132� I

-xx41-x24-x4

x Z 0,- ƒ -6 ƒ - 82 + 1-1249 # 48 , 1-42 # 3 + 1-523-263 + 264

-241-225-23-52 + 31-222-321-32210 - 13 - 12210 - 3 - 12

a = -2, b = -3, c = 4a = -5, b = 2, c = -6

a1b + c2 = ab + ac,

20°C.

5. Use the definition of division to find each quotient, ifpossible. If a quotient is not defined, explain why.a. b.c.

6. Evaluate each of the following, if possible:a. b.c. d.

7. Evaluate each of the following products and then, if pos-sible, write two division statements that are equivalent toyour multiplication statements. If two division state-ments are not possible, explain why.a.b.

8. Suppose a and b are integers and is an integer.a. Use the definition of integer division to prove that if

then b. Why is the statement in (a) not true if Is it

true if and Justify your answers.9. In each of the following, x and y are integers. Use the

definition of division in terms of multiplication to per-form the indicated operations. Write your answers insimplest form.a.b.

10. In a lab, the temperature of various chemical reactionswas changing by a fixed number of degrees per minute.Write a numeric expression that describes each of thefollowing:a. The temperature at 8:00 A.M. is If it increases

by d degrees per minute, what will the temperaturebe m minutes later?

b. The temperature at 8:00 P.M. is If it dropped ddegrees per minute, what was the temperature mminutes before?

0°C.

-5°C.

1-10x + 52 , 51-4x2 , x

c Z 0?a , b = 0c = 0?

1ac2 , 1bc2 = a , b.c Z 0,

a , b1-421-321-524

1-23 - -72 , 41-8 + 82 , 81ab2 , b1a , b2b

0 , 00 , 1-52143 , 1-112

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b.c.d.e. is negative.f.g.h.

21. Solve the following for x or find the values of the indi-cated expression:a.b.c.d.e. If find the values of f. If find the values of

22. Use the difference-of-squares formula to simplify eachof the following, if possible:a.b.c.

23. Factor each of the following expressions completely andthen simplify, if possible:a. b.c. d.e.f.g.

24. If x and y are integers, classify each of the following astrue or false. If true, explain why. If false, give a counter-example.a.b.c.d.

25. a. Given a calendar for any month of the year, such asthe one that follows, pick several groups ofnumbers and find the sum of these numbers. How arethe sums obtained related to the middle number?

3 * 3

ƒ x ƒ 2 = x2ƒ x2 ƒ = x2ƒ xy ƒ = ƒ x ƒ ƒ y ƒƒ x + y ƒ = ƒ x ƒ + ƒ y ƒ

1x2 - y22 + x + yx2 - 9y21a + b21c + 12 - 1a + b2

3x2 + xy - x3x - 4x + 7xax - 2xax + 2x

2132 - 1321x - 1211 + x212 + 3x212 - 3x2

2 - 7x.x 6 0,3 - 5x.x 7 -2,

-51x - 32 7 -5-6x 7 -x + 20x3 = -2912x - 122 = 11 - 2x22

-31x + 22 = -3x + 6x - 3x = -2x-11 - x2 = x - 1-x2-x , -x = 1x2 = -9x2 = 9

S M T W T F S

1

8

15

22

29

17

2

9

16

23

30

3

10

17

24

31

4

11

18

25

6

13

20

27

7

14

21

28

5

12

19

26

JULY

b. Prove that the sum of any nine digits in any setof numbers selected from a monthly calendar will al-ways be equal to 9 times the middle number.

26. In each of the following, find the next two terms. As-sume the sequence is arithmetic or geometric, and findits difference or ratio and the nth term.a.b.

27. Find the sum of the first 100 terms of the arithmeticsequence

28. Find the first five terms of the sequences whose nth termisa.b.c.

29. In the geometric sequence determine ifthere is a term equal to the following numbers:a. 512 b. 1024

30. If x and y are integers, classify each of the following astrue or false. Justify your answers.a. b.c. If then for all integers a.

31. Jon has two checking accounts. In the first one, he is$120 overdrawn, and in the second, his balance is $300.If he deposits $40 every day in the first account but with-draws $20 daily from the second account, after howmany days will the balance in each account be the same?Explain your solution.

a - x 6 a - y,x 6 yƒ x ƒ 7 -11-x23 = -x3

1, - 2, 4, - 8, Á ,ƒ 10 - n2 ƒn21-12n1-22n + 2n

10, 7, 4, 1, -2, -5, Á .

-2, 4, -8, 16, -32, 64, _, _10, 7, 4, 1, -2, -5, _, _

3 * 3

Communication1. Can be multiplied by using the

difference-of-squares formula? Explain why or whynot.

2. Kahlil said that using the equation he can find a similar equation for

Examine his argument. If it is correct, supply any miss-ing steps or justifications; if it is incorrect, point out why.

= a2 - 2ab + b2 = a2 + 2a1-b2 + 1-b22

1a - b22 = 3a + 1-b242

1a - b22.2ab + b2,1a + b22 = a2 +

1-x - y21x + y23. a. Use the distributive property of multiplication over

addition to show that (Hint: Write)

b. Use part (a) to show that 4. Nancy gave the following argument to show that

for all integers a and b: I know thathence:

If the argument is valid, complete its details; if it is notvalid, explain why not.

= -1ab2 = 1-121ab2

1-a2b = 31-12a4b

1-12a = -a;1-a2b = -1ab2,

1-12a = -a.a as 1 # a.

1-12a + a = 0.


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FLOUR

+4

FLOUR

–3

FLOUR

+5

FLOUR

–6

5. Hosni gave the following argument that for all integers a and b. If the argument is cor-

rect, supply the missing reasons. If it is incorrect, explainwhy.

6. The Swiss mathematician Leonhard Euler (1707–1783)argued that as follows: “The result mustbe either or 1. If it is then Because we have Now dividing both sides of the last equation by weget which of course cannot be true. Hence

must be equal to 1.”a. What is your reaction to this argument? Is it logical?

Why or why not?b. Can Euler’s approach be used to justify other proper-

ties of integers? Explain.7. If answer the following:

a. Find the greatest integer x for which the inequality istrue. Explain your thinking.

b. Is there a least integer x for which the inequality istrue? Explain why or why not.

8. Jill asks each of her classmates to choose a number, thenmultiply the number by add 2 to the product, multi-ply the result by and then subtract 14. Finally, eachstudent is asked to divide the result by 6 and record theanswer. When Jill gets an answer from a classmate, shejust adds 3 to it in her head and announces the numberthat classmate originally chose. How did Jill know to add3 to each answer?

Open-Ended9. Make up a problem similar to problem 8 but with all

numbers different and solve it.10. On a national mathematics competition, scoring is ac-

complished using the formula 4 times the number donecorrectly minus the number done incorrectly. In thisscheme, problems left blank are considered neither cor-rect nor incorrect. Devise a scenario that would allow astudent to have a negative score.

11. Select a current middle-school text that introduces mul-tiplication and division of integers and discuss any mod-els that were used and how effective you think theywould be with a group of students.

Cooperative Learning12. Devise a scheme for determining a grade-point average

for a college student that allows negative quality pointsfor a failing grade.a. Use your scheme to determine possible grades for

students with positive, zero, and negative grade-pointaverages.

b. Compare your scheme with that of another class groupand write a rationale for the best scheme.

13. a. How would you introduce multiplication of integersin a middle-school class and how would you explain

-2,-3,

5x + 3 6 -20,

1-121-12-1 = 1,

-11-121-12 = 1-121.-1 = 1-121,1-121-12 = -1.-1,-1

1-121-12 = 1

= -a + -b = 1-12a + 1-12b

- 1a + b2 = 1-121a + b2

-a + -b,-1a + b2 = that a product of two negative numbers is positive?

Write a rationale for your approach.b. Present your answers and compare them to those of

another class group and together decide the mostappropriate way to introduce the concepts. Write arationale for your approach.

14. Discuss in your group what each person’s favorite ap-proach is to justify the fact that and whyit is the favorite.

Questions from the Classroom15. A seventh-grade student does not believe that

The student argues that a debt of $5 is greater than adebt of $2. How do you respond?

16. A student computes by writing How would you help this student?

17. A student says that his father showed him a very simplemethod for dealing with expressions like and The rule is, if there is a negative signbefore the parentheses, change the signs of the expres-sions inside the parentheses. Thus,

and What is your response?

18. Al said that 4 and can’t both be He saidhe knows that because

Therefore, must be What would you tell Al?

19. Betty used the charged-field model to show thatShe said that this proves that any negative

integer times a negative integer is a positive integer.How do you respond?

Review Problems20. Illustrate on a number line.21. Find the additive inverse of each of the following:

a. b. 7 c. 022. Compute each of the following:

a.b.c.d.

23. In the 1400s, European merchants used positive andnegative numbers to label barrels of flour. For example,a barrel labeled meant the barrel was 3 lb over-weight, whereas a barrel labeled meant the barrelwas 5 lb underweight. If the following numbers werefound on 100 lb barrels, what was the total weight of thebarrels?

-5+3

ƒ 11 ƒ + ƒ -11 ƒ8 - ƒ -12 ƒƒ -14 ƒ + 7ƒ -14 ƒ

-5

-8 + -5

-21-32 = 6.

8.-41-22+ -2 = -8.-2 + -2-2 +41-22 =41-22 = -8

8.-41-221-22

x - 2x + 3.x - 12x - 32 =-a + b - 1-1a - b + 12 =

x - 12x - 32.-1a - b + 12

= 30.

-101-32-8 - 21-32

-5 6 -2.

1-121-12 = 1

24. Write the function without ab-solute value. (Distinguish two cases: and )x 6 0.x Ú 0

f 1x2 = 1x + ƒ x ƒ 2 , 2

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5-3 Divisibility

The concepts of even and odd integers are commonly used. For example, during water short-ages in the summer in some parts of the country, houses with even-number addresses canwater on even-numbered days of the month and houses with odd-number addresses canwater on odd-numbered days. An even integer is an integer that is divisible by 2; that is, ithas 0 remainder when divided by 2. An odd integer is an integer that is not divisible by 2.The fact that 12 is divisible by 2 can be stated in the following equivalent statements in theleft column:

Definition of “Divides”If a and b are any integers, then b divides a, written , if, and only if, there is a unique integer qsuch that .a = bq

b ƒ a

Example General Statement12 is divisible by 2. a is divisible by b.2 is a divisor of 12. b is a divisor of a.12 is a multiple of 2. a is a multiple of b.2 is a factor of 12. b is a factor of a.2 divides 12. b divides a.

The statement that “2 divides 12” is written with a vertical segment, as in where thevertical segment means divides. Likewise, “b divides a” can be written . Each statementin the right column above can be written . We write to symbolize that 5 does notdivide 12 or that 12 is not divisible by 5. The notation also implies that 12 is not amultiple of 5 and 5 is not a factor of 12.

In general, if a is a nonnegative integer and b is a positive integer, we say that a is divisi-ble by b or equivalently that b divides a if, and only if, the remainder when a is divided by bis 0. Using the division algorithm, this means that there is a unique q (quotient) such that

. We extend this concept to divisibility for all integers in the following definition.a = bq

5�125�12b ƒ a

b ƒ a2 ƒ 12,

Section 5-3 Divisibility 285

25. Solve for x if possible:a. b.c. d.

Third International Mathematics and Science Study(TIMSS) QuestionsIf n is a negative integer, which of these is the largest number?

a. b. c. d. 3 , n3 - n3 * n3 + n

ƒ x ƒ = -xƒ x ƒ = xƒ x ƒ + 1 = 0ƒ x ƒ = 3

If what is the value of a.b.c.d. 1e. 9TIMSS, Grade 8, 2003

-1-6-9

-3x?x = -3,

BRAIN TEASER If are consecutive letters of the alphabet representing integers, find the product:

1x - a21x - b21x - c2 # Á # 1x - z2

a, Á , z

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Do not confuse with , which is interpreted as . The former, a relation, iseither true or false. The latter, an operation, has a numerical value if . Note that if is an integer, then . Also note that for positive integers is equivalent to saying thatthe remainder when b is divided by a is not 0.

a�ba ƒ bb>aa Z 0

b , ab>ab ƒ a

▲▲


Pierre de Fermat (1601–1665) was a lawyer and a magistrate who served in the provincialparliament in Toulouse, France. He devoted his leisure time to mathematics—a subject inwhich he had no formal training. After his death, his son decided to publish a new edition ofDiophantus’s Arithmetica with Fermat’s notes. One of the notes in the margin of Fermat’scopy asserted that the equation has no positive-integer solutions if n is an inte-ger greater than 2 and commented, “I have found an admirable proof of this, but the marginis too narrow to contain it.” Many great mathematicians spent years trying to prove Fermat’sassertion, now called “Fermat’s last theorem.” In 1995, Andrew Wiles, a Princeton Univer-sity mathematician, proved Fermat’s last theorem.

xn + yn = zn

Classify each of the following as true or false. Explain your answers.

a. b. c. 0 is even.d. e. For all integers . f. For all integers g. for all integers n. h. if a and b are integers and i.

Solution a. is true because b. is false because there is no integer c such that c. is true because therefore, 0 is even.d. is true because there is no integer c such that e. is true for all integers a because f. is true for all integers a because g. is true. Because is a multiple of 3 and hence h. is true because and i. is false because for all integers q, so q is not unique.0 = 0 # q0 ƒ 0

a Z b.a2 - b2 = 1a - b21a + b21a - b2 ƒ 1a2 - b22

3 ƒ 6n.6n = 3 # 2n, 6n3 ƒ 6na = 1-a21-12.-1 ƒ a

a = a # 1.1 ƒ a2 = c # 8.8�2

0 = 0 # 2;2 ƒ 02 = c # 0.0 ƒ 2

12 = -41-32.-3 ƒ 12

0 ƒ 0a Z b.1a - b2 ƒ 1a2 - b223 ƒ 6n

a, -1 ƒ a.a, 1 ƒ a8�20 ƒ 2-3 ƒ 12▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-16

▲

▲

Suppose we have one pack of gum and we know that the number of pieces, a, is divisibleby 5. Then if we had two packs, the total number of pieces of gum is still divisible by 5. Thesame is true if we had 10 packs, or 100 packs or in general n packs where n is any positiveinteger. We could record this observation as follows:

Even more generally, if then d divides any multiple of a. We state this fact in the follow-ing theorem.

d ƒ a

If 5 ƒ a, then 5 ƒ na, where a and n are integers and n 7 0.

Theorem 5–12For any integers a and d, if and n is any integer, then .d ƒ nad ƒ a

R E M A R K In this chapter whenever the “divides” symbol is used as in we assumethat and are integers and b Z 0.ba

b ƒ a,

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The theorem can be stated in an equivalent form:If d is a factor of a (that is, a equals some integer times d), then d is a factor of any multiple of a.

Next consider two different brands of chewing gum each having five pieces, as inFigure 5-16. We can divide each package of gum evenly among five students. In addi-tion, if we opened both packages and put all of the pieces in a bag, we could still dividethe number of pieces of gum evenly among the five students. To generalize this notion,if we buy gum in larger packages with a pieces in one package and b pieces in a secondpackage with both a and b divisible by 5, we can record the preceding discussion asfollows:

If the number, a, of pieces of gum in one package is divisible by 5, but the number, b, ofpieces in the other package is not, then the total, , cannot be divided evenly amongthe five students. This can be recorded as follows:

If 5 ƒ a and 5�b, then 5�1a + b2.

a + b

If 5 ƒ a and 5 ƒ b, then 5 ƒ 1a + b2.

NOW TRY THIS 5-8 If what, if anything, is wrong with the statement: if and then5�1a + b2?

5�b,5�aa, b � I,

Theorem 5–13For any integers a, b, and d,

a. If and then b. If and then c. If and then d. If and then d�1a - b2.d�b,d ƒ a

d ƒ (a - b).d ƒ b,d ƒ ad�1a + b2.d�b,d ƒ a

d ƒ 1a + b2.d ƒ b,d ƒ a

d Z 0,

Spear Bubble

Chewdent

Figure 5-16

Since subtraction can be defined in terms of addition, results similar to addition hold forsubtraction. These ideas are generalized in Theorem 5–13.

R E M A R K Theorem 5–13 can be extended. For example, if a, b, c, and d are integers,, then,

If d ƒ a, d ƒ b and d ƒ c, then d ƒ 1a + b + c2.

d Z 0

The proofs of most theorems in this section are left as exercises, but the proof ofTheorem 5–13(a) is given as an illustration.

Proof

Theorem 5–13(a) is equivalent to the following:

Notice that “a is a multiple of d ” means for some integer m. Similarly “b is a mul-tiple of d” means for some integer n. To show that is a multiple of d, we addthese equations as follows:

a + b = md + nd

a + bb = nd,a = md,

If a is a multiple of d and b is a multiple of d, then a + b is a multiple of d.

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Is a multiple of d? Notice that so Bythe closure property of integer addition, is an integer. Therefore, is a multipleof d and therefore d ƒ 1a + b2.

a + bm + na + b = 1m + n2d.md + nd = 1m + n2d,md + nd

Classify each of the following as true or false, where x, y, and z are integers. If a statementis true, prove it. If a statement is false, provide a counterexample.

a. If and then b. If then and c. If then

Solution a. True. By Theorem 5–12, if then, for any integer or b. False; for example, but and .c. False; for example, but 3 ƒ 21.9�21,

3�23�73 ƒ 17 + 22,3 ƒ xy.y, 3 ƒ yx3 ƒ x,

3�a.9�a,3 ƒ y.3 ƒ x3 ƒ 1x + y2,3 ƒ xy.3 ƒ y,3 ƒ x

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼Example 5-17

▲

▲

NOW TRY THIS 5-9 If and if is it true that regardless of whether or Why?3�y?3 ƒ y3 ƒ xy3 ƒ x,x, y � I,

Divisibility RulesAs shown in Example 5-18, sometimes it is handy to know if one number is divisible by an-other just by looking at it or by performing a simple test. We discovered that if a numberends in 0, then the number is divisible by 5. The same argument can be used to show that ifa number ends in 5, it is divisible by 5. This is an example of a divisibility rule. Moreover, ifthe last digit of a number is neither 0 nor 5, then the number is not divisible by 5.

Elementary texts frequently state divisibility rules. However, such rules have limited useexcept for mental arithmetic. It is possible to determine whether 1734 is divisible by 17, ei-ther by using pencil and paper or a calculator. To check divisibility and avoid decimals, wecan use a calculator with an integer division button, . On such a calculator, integerdivision can be performed using the following sequence of buttons:

to obtain the display .

This implies with a remainder of 0, which, in turn, implies 17 ƒ 1734.1734>17 = 102

102 0

=71INT ,4371

INT ,

Five students found a padlocked money box that had a deposit slip attached to it. The depositslip was water-spotted, so the currency total appeared as shown in Figure 5-17. One studentremarked that if the money listed on the deposit slip was in the box, it could easily be dividedequally among the five students without using coins. How did the student know this?

Solution Because the units digit of the amount of the currency is 0, the solution to theproblem is to determine whether all natural numbers whose units digit is 0 are divisible by 5.To solve this problem, look for a pattern. Natural numbers whose units digit is 0 form a pat-tern, that is, 10, 20, 30, 40, 50, . These numbers are multiples of 10. We are to determinewhether 5 divides all multiples of 10. Since by Theorem 5–12, 5 divides any multipleof 10. Hence, 5 divides the amount of money in the box, and the student is correct.

5 ƒ 10,Á

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-18

▲

▲

$ 0.00

Figure 5-17

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We could have determined this same result mentally by considering the following:

Because and by Theorem 5–13(a), we have or Similarly, we could determine mentally that 17�1735.

17 ƒ 1734.17 ƒ 11700 + 342,17 ƒ 34,17 ƒ 1700

1734 = 1700 + 34

R E M A R K Notice that implies that is, In general, if, and only if, .d ƒ ad ƒ -a

17 ƒ -1734.17 ƒ 1-121734,17 ƒ 1734

Divisibility Tests for 2, 5, and 10

To determine mentally whether a given integer n is divisible by another integer d, we thinkof n as the sum or difference of integers, where d divides at least one of the numbers. Weuse a concrete example to get an idea of how this concept works. Consider the number1362. This number can be represented as in Figure 5-18. Notice that 2 divides each part ofthe figure (see dashed lines). Since 2 divides each part of the figure, by the extension ofTheorem 5–13, 2 divides the sum of all of the parts and hence 2 ƒ 1362.

Figure 5-18

1362

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Theorem 5–17: Divisibility Test for 4An integer is divisible by 4 if, and only if, the last two digits of the integer represent a numberdivisible by 4.

Theorem 5–14: Divisibility Test for 2An integer is divisible by 2 if, and only if, its units digit is divisible by 2.

Theorem 5–15: Divisibility Test for 5An integer is divisible by 5 if, and only if, its units digit is divisible by 5; that is, if, and only if,the units digit is 0 or 5.

Theorem 5–16: Divisibility Test for 10An integer is divisible by 10 if, and only if, its units digit is divisible by 10; that is, if, and only if,the units digit is 0.

Divisibility Tests for 4 and 8

When we consider divisibility rules for 4 and 8, we see that and , so it is not amatter of checking the units digit for divisibility by 4 and 8. However, 4 divides and 8 divides

We first develop a divisibility rule for 4. Consider any four-digit number n such thatOur subgoal is to write the given number as a sum of two

numbers, one of which is as great as possible and divisible by 4. We know that becauseand, consequently, Because then and because then

Finally, and imply . Now the divisibility ofby 4 depends on the divisibility of by 4. Notice that

is the number represented by the last two digits in the given number n. Wesummarize this in the following test.c10 + d

1c10 + d2a103 + b102 + c10 + d4 ƒ 1a103 + b10224 ƒ b1024 ƒ a1034 ƒ a103.

4 ƒ 103,4 ƒ b102;4 ƒ 102,4 ƒ 103.102 = 4 # 254 ƒ 102

n = a103 + b102 + c10 + d.

103.1which is 232102,1which is 222

8�104�10

Note that if the original number was 1363, then and , so . We seethat all we have to do is to determine whether the units digit is divisible by 2 in order todetermine if the number is divisible by 2. We can develop a similar test for divisibilityby 5 and 10.

2�13632�12 ƒ 1362

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a. Determine whether 97,128 is divisible by 2, 4, and 8.b. Determine whether 83,026 is divisible by 2, 4, and 8.

Solution a. because because because

b. because because because 8�026.8�83,026

4�26.4�83,0262 ƒ 6.2 ƒ 83,0268 ƒ 128.8 ƒ 97,1284 ƒ 28.4 ƒ 97,1282 ƒ 8.2 ƒ 97,128

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-19

▲

▲

Theorem 5–18: Divisibility Test for 8An integer is divisible by 8 if, and only if, the last three digits of the integer represent a numberdivisible by 8.

Use Theorems 5–12 and 5–13 to show why the divisibility test for 8 works in Example 5-19(b).

Solution We can write 97,128 as . Because , then or(Theorem 5–12). Next we need to check if . It does, so or(Theorem 5–13).8 ƒ 97,128

8 ƒ 197,000 + 12828 ƒ 1288 ƒ 97,0008 ƒ 97 # 10008 ƒ 100097,000 + 128

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-20

▲

▲

To investigate divisibility by 8, we note that the least positive power of 10 divisible by 8 issince Consequently, all integral powers of 10 greater than also are

divisible by 8. Hence, the following is a divisibility test for 8.103103 = 8 # 125.103

INumbers divisible

by 2

Numbers divisibleby 4

Numbers divisibleby 8

Figure 5-19

Notice that if we cannot conclude that or Why?2�a.4�a8�a,

R E M A R K In Example 5–19(a), it would have been sufficient to check that the givennumber is divisible by 8 because if then and Why? This relationship isshown in Figure 5-19.

4 ƒ a.2 ƒ a8 ƒ a,

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Next, we consider a divisibility test for 3. No power of 10 is divisible by 3, but the numbers9, and 99, and 999, and others of this type are divisible by 3. For example, to determinewhether 5721 is divisible by 3, we rewrite the number using 999, 99, and 9, as follows:

The sum in the first set of parentheses in the last line is divisible by 3, so the divisibilityof 5721 by 3 depends on the sum in the second set of parentheses. In this case,

and so Hence, to test 5721 for divisibility by 3, we test for divisibility by 3. Notice that is the sum of the digits of 5721. The

example suggests the following test for divisibility by 3.5 + 7 + 2 + 12 + 1

5 + 7 +3 ƒ 5721.3 ƒ 15,7 + 2 + 1 = 155 +

= 15 # 999 + 7 # 99 + 2 # 92 + 15 + 7 + 2 + 12 = 5 # 999 + 5 # 1 + 7 # 99 + 7 # 1 + 2 # 9 + 2 + 1 = 51999 + 12 + 7199 + 12 + 219 + 12 + 1

5721 = 5 # 103 + 7 # 102 + 2 # 10 + 1

Theorem 5–19: Divisibility Test for 3An integer is divisible by 3 if, and only if, the sum of its digits is divisible by 3.

We can use an argument similar to the one used to demonstrate that to prove thetest for divisibility by 3 on any integer and in particular for any four-digit number

Even though is not necessarily divis-ible by 3, the number is divisible by 3. We have the following:

Because and it follows that If then that is, If, on the other hand,

it follows from Theorem 5–13(b) that Since and so on, a test similar to that for divisibility by 3 applies to

divisibility by 9. Why?9 ƒ 9, 9 ƒ 99, 9 ƒ 999,

3�n.3�1a + b + c + d2,3 ƒ n.3 ƒ 31a999 + b99 + c92 + 1a + b + c + d24;

3 ƒ 1a + b + c + d2,3 ƒ 1a999 + b99 + c92.3 ƒ 999,3 ƒ 9, 3 ƒ 99,

= 1a999 + b99 + c92 + 1a + b + c + d2 = 1a999 + b99 + c92 + 1a1 + b1 + c1 + d2 = a1999 + 12 + b199 + 12 + c19 + 12 + d

a103 + b102 + c10 + d = a1000 + b100 + c10 + d

a999 + b99 + c9a103 + b102 + c10 + dn = a103 + b102 + c10 + d.

3 ƒ 5721

Use divisibility tests to determine whether each of the following numbers is divisible by 3and divisible by 9:

a. 1002 b. 14,238

Solution a. Because and it follows that Because itfollows that 9�1002.

9�3,3 ƒ 1002.3 ƒ 3,1 + 0 + 0 + 2 = 3

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-21

Theorem 5–20: Divisibility Test for 9An integer is divisible by 9 if, and only if, the sum of the digits of the integer is divisible by 9.

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The divisibility test for 7 is usually harder to use than actually performing the division, sowe omit the test. We state the divisibility test for 11 but omit the proof. Interested readersmight try to find a proof.

The store manager has an invoice for 72 four-function calculators. The first and last digitson the receipt are illegible. The manager can read

What are the missing digits, and what is the cost of each calculator?

Solution Let the missing digits be x and y so that the number is dollars, or cents. Because there were 72 calculators sold, the number on the invoice must be divisibleby 72. Because the number is divisible by 72 and it must be divisible by 8 and 9,which are factors of 72. For the number on the invoice to be divisible by 8, the three-digitnumber must be divisible by 8. Because must be divisible by 8, it is an even number.Therefore, must be either 790, 792, 794, 796, or 798. Only the number 792 is divisibleby 8, so we know the last digit, y, on the invoice must be 2.

Because the number on the invoice must be divisible by 9, we know that 9 must divideor Since 3 is the only single digit that will make

divisible by 9, then x must be 3. Therefore, the number on the invoice must be $367.92.The calculators must cost , or $5.11, each.$367.92>72

1x + 2421x + 242.x + 6 + 7 + 9 + 2,

79y79y79y

72 = 8 # 9,

x679yx67.9y

$ . 67.9 .

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼Example 5-22

▲

▲

Theorem 5–21: Divisibility Test for 11An integer is divisible by 11 if, and only if, the sum of the digits in the places that are even powersof 10 minus the sum of the digits in the places that are odd powers of 10 is divisible by 11.

For example, to test whether 8,471,986 is divisible by 11, we check whether 11 divides thedifference or 17. Because it follows from the di-visibility test for 11 that A number like 2772 is divisible by 11 because

and 0 is divisible by 11.The divisibility test for 6 is related to the divisibility tests for 2 and 3. In Section 5-4, we

will ask you to show that if and then and in general: if a and c have nofactors in common, then if and , we can conclude that . Consequently, the follow-ing divisibility test is true.

ac ƒ bc ƒ ba ƒ b12 # 32 ƒ n,3 ƒ n,2 ƒ n

(2 + 7) - (7 + 2) = 9 - 9 = 011�8,471,986.

11�17,16 + 9 + 7 + 82 - 18 + 1 + 42,

Theorem 5–22: Divisibility Test for 6An integer is divisible by 6 if, and only if, the integer is divisible by both 2 and 3.

Divisibility tests for other numbers are explored in Assessments 5-3A and 5-3B.

b. Because and it follows that Becauseit follows that 9 ƒ 14,238.9 ƒ 18,

3 ƒ 14,238.3 ƒ 18,1 + 4 + 2 + 3 + 8 = 18

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A 20th century mathematician who worked in the area of number theory was American JuliaRobinson (1919–1985). Robinson was the first woman mathematician to be elected to theNational Academy of Sciences and the first woman president of the American MathematicalSociety. She died of leukemia at the age of 65. ▲▲

The number 57,729,364,583 has too many digits for most calculator displays. Determinewhether it is divisible by each of the following:

a. 2 b. 3 c. 5 d. 6 e. 8 f. 9 g. 10 h. 11

Solution a. No, the units digit, 3, is not divisible by 2.b. No, the sum of the digits is 59, which is not divisible by 3.c. No, the units digit is neither 0 nor 5.d. No, the number is not divisible by 2.e. No, the number formed by the last three digits, 583, is not divisible by 8.f. No, the sum of the digits is 59, which is not divisible by 9.g. No, the units digit is not 0.h. Yes,

and 11 is divisible by 11.13 + 5 + 6 + 9 + 7 + 52 - 18 + 4 + 3 + 2 + 72 = 35 - 24 = 11

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-23

▲

▲

NOW TRY THIS 5-10 Fill in the following blanks so that the number is divisible by 9. List allpossibilities.

12,506,5_ _.

Problem Solving A Mistake in the Inventory

A class from Washington School visited a neighborhood cannery warehouse. The warehousemanager told the class that there were 11,368 cans of juice in the inventory and that the canswere packed in boxes of 6 or 24, depending on the size of the can. One of the students, Sam,thought for a moment and announced that there was a mistake in the inventory. Is Sam’sstatement correct? Why or why not?

Understanding the Problem The problem is to determine if the manager’s inventory of11,368 cans was correct. To solve the problem, we must assume there are no partial boxesof cans; that is, a box must contain exactly 6 or exactly 24 cans of juice.

Devising a Plan We know that the boxes contain either 6 cans or 24 cans, but we do notknow how many boxes of each type there are. One strategy for solving this problem is tofind an equation that involves the total number of cans in all the boxes.

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The total number of cans, 11,368, equals the number of cans in all the 6-can boxes plusthe number of cans in all the 24-can boxes. If there are n boxes containing 6 cans each,there are 6n cans altogether in those boxes. Similarly, if there are m boxes with 24 canseach, those boxes contain a total of 24m cans. Because the total was reported to be 11,368cans, we have the equation Sam claimed that

One way to show that is to show that and 11,368 do nothave the same divisors. Both and are divisible by 6. This implies that must be divisible by 6. If 11,368 is not divisible by 6, then Sam is correct.

Carrying Out the Plan The divisibility test for 6 states that a number is divisible by 6 if, andonly if, the number is divisible by both 2 and 3. Because 11,368 ends in 8, it is divisible by2. Is it divisible by 3?

The divisibility test for 3 states that a number is divisible by 3 if, and only if, the sum ofthe digits in the number is divisible by 3. We see that which isnot divisible by 3, so 11,368 is not divisible by 3. Hence, Sam is correct.

Looking Back Suppose 11,368 had been divisible by 6. Would that have implied that themanager was correct? The answer is no; it would have implied only that we would have tochange our approach to the problem.

As a further Looking Back activity, suppose that, given different data, the manager is cor-rect. Can we determine values for m and n? In fact, this can be done. If a computer is avail-able, a program can be written to determine all possible natural-number values of m and n.

1 + 1 + 3 + 6 + 8 = 19,

6n + 24m24m6n6n + 24m6n + 24m Z 11,368

6n + 24m Z 11,368.6n + 24m = 11,368.

▲▲BRAIN TEASER The following is an argument to show that an ant weighs as much as an elephant.What is wrong?

Let e be the weight of the elephant and a the weight of the ant. Let . Consequently,. Multiply each side of by . Then simplify.

Thus, the weight of the elephant equals the weight of the ant.

e = a e(e - a - d) = a(e - a - d)

e2 - ea - de = ae - a2 - da e2 - ea = ae + de - a2 - da e(e - a) = (a + d)(e - a)

e - ae = a + de = a + de - a = d

Assessment 5-3A

1. Classify each of the following as true or false. If false, tellwhy.a. 6 is a factor of 30. b. 6 is a divisor of 30.c. d. 30 is divisible by 6.e. 30 is a multiple of 6. f. 6 is a multiple of 30.

6 ƒ 30.

2. Using divisibility tests, answer each of the following:a. There are 1379 children signed up to play in a base-

ball league. If exactly 9 players are to be placed oneach team, will any team be short of players?

b. A forester has 43,682 seedlings to be planted. Canthese be planted in an equal number of rows with11 seedlings in each row?IS

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c. There are 261 students to be assigned to 9 teachersso that each teacher has the same number of students.Is this possible?

3. Without using a calculator, test each of the followingnumbers for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11:a. 746,988b. 81,342c. 15,810

4. Determine each of the following without actually per-forming the division. Explain how you did it in each case.a. Is 34,015 divisible by 17?b. Is 34,051 divisible by 17?c. Is 19,031 divisible by 19?d. Is divisible by 5?e. Is divisible by 5?

5. Justify each of the given statements, assuming that a, b,and c are integers. If a statement cannot be justified byone of the theorems in this section, answer “none.”a. implies b. and imply .c. and imply .d. and imply e. implies

6. Classify each of the following as true or false. Justifyyour answers.a. If , then .b. If , then c. If , then and

7. Justify each of the following:a.b.c.d.

8. Classify each of the following as true or false:a. If every digit of a number is divisible by 3, the number

itself is divisible by 3.b. If a number is divisible by 3, then every digit of the

number is divisible by 3.9. Fill each of the following blanks with the greatest digit

that makes the statement true:a.b.c.

10. Place a digit in the square, if possible, so that thenumber

is divisible bya. 2 b. 3c. 4 d. 9e. 11

527,4 n 2

11 ƒ 6_559 ƒ 83_453 ƒ 74_

7�14200 + 2226 ƒ 23 # 32 # 17419 ƒ 11900 + 3827 ƒ 210

-b ƒ -a.b ƒ -ab ƒ ab2 ƒ a3.b ƒ a1b + c2 ƒ 1a + c2b ƒ a

3 ƒ a2.3 ƒ a3�1a + b + c2.3�c3 ƒ (a + b)

4�13004�134 ƒ 1004�1100 + 1324�134 ƒ 100

4 ƒ 113 # 20.4 ƒ 20

12 # 3 # 5 # 72 + 12 # 3 # 5 # 7

11. The bookstore marked some notepads down from $2.00but still kept the price over $1.00. It sold all of them.The total amount of money from the sale of the padswas $31.45. How many notepads were sold?

12. A group of people ordered pencils. The bill was $2.09. Ifthe original price of each was 12¢ and the price hasrisen, how much does each cost?

13. Leap years occur in years that are divisible by 4. How-ever, if the year ends in two zeros, in order for the yearto be a leap year, it must be divisible by 400. Determinewhich of the following are leap years:a. 1776b. 1986c. 2000d. 2100

14. A test for checking computations is called casting outnines. Consider the sum Theremainders when 193, 24, and 786 are divided by 9 are4, 6, and 3, respectively. The sum of the remainders, 13,has a remainder of 4 when divided by 9, as does 1003.Checking the remainders in this manner provides aquasi-check for the computation. Find the followingsums and use casting out nines to check your sums:a.b.c.d. Try the check on the subtraction e. Try the check on the multiplication f. Would it make sense to try the check on division?

Why or why not?15. a. If 21 divides n, what other natural numbers divide n?

Why?b. If 16 divides n, what other natural numbers divide n?

Why?16. The numbers x and y are divisible by 5.

a. Is the sum of x and y divisible by 5? Why?b. Is the difference of x and y divisible by 5? Why?c. Is the product of x and y divisible by 5? Why?

17. Using only divisibility tests, explain whether 6,868,395is divisible by 15.

18. Classify each of the following as true or false, assumingthat a, b, c, and d are integers. If a statement is false, givea counterexample.a. If then and .b. If then or .c. If , then or .d. If then and .e. If and , then .

19. Prove the test for divisibility by 9 for any five-digitnumber.

a = bb ƒ aa ƒ bb ƒ ca ƒ cab ƒ c,

d ƒ bd ƒ ad ƒ abd ƒ bd ƒ ad ƒ (a + b),

d ƒ bd ƒ ad ƒ (a + b),

345 # 56.1003 - 46.

10,034 + 3004 + 400 + 20987 + 456 + 876512,343 + 4546 + 56

193 + 24 + 786 = 1003.

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Assessment 5-3B

1. Classify each of the following as true or false. If false, tellwhy.a. 5 is a multiple of 20.b. 10 is a divisor of 30.c.d. 10 is divisible by 1.e. 30 is a factor of 6.f. 6 is a multiple of 20.

2. Using divisibility tests, answer each of the following:a. Six friends win with a lottery ticket. The payoff is

$242,800. Can the money be divided evenly?b. Jack owes $7812 on a new car. Can this amount be

paid in 12 equal monthly installments?3. Without using a calculator, test each of the following

numbers for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11:a. 4,201,012 b. 1001c. 10,001

4. Determine each of the following without actually per-forming the division. Explain how you did it in each case.a. Is 24,013 divisible by 12?b. Is 24,036 divisible by 12?c. Is 17,034 divisible by 17?d. Is divisible by 3?e. Is divisible by 6?

5. Justify each of the following.a. if b. if c. if if d. If and then

6. Justify each of the following:a.b.c.d.

7. Classify each of the following as true or false:a. If a number is divisible by 6, then it is divisible by 2

and by 3.b. If a number is divisible by 2 and 3, then it is divisible

by 6.c. If a number is divisible by 2 and 4, then it is divisible

by 8.d. If a number is divisible by 8, then it is divisible by

2 and 4.8. Devise a test for divisibility by each of the following

numbers:a. 16b. 25

24 ƒ 1104 + 642

24�12 # 4 # 6 # 8 # 1710 + 1213�124 # 53 # 26 + 1226 ƒ 1134 # 1002

bc ƒ ac.c Z 0,b ƒ aa Z 0.0 … n … m,an ƒ am,

a Z 0.a4 ƒ a10,a Z 0.a3 ƒ a4,

12 # 3 # 5 # 72 + 12 # 3 # 5 # 7

8 ƒ 32.

9. When the two missing digits in the following numberare replaced, the number is divisible by 99. What is thenumber?

10. Without using a calculator, classify each of the followingas true or false. Justify your answers.a.b.c.d.

11. In a football game, a touchdown with an extra point isworth 7 points and a field goal is worth 3 points. Sup-pose that in a game the only scoring done by teams aretouchdowns with extra points and field goals.a. Which of the scores 1 to 25 are impossible for a team

to score?b. List all possible ways for a team to score 40 points.c. A team scored 57 points with 6 touchdowns and 6 extra

points. How many field goals did the team score?12. Complete the following table where n is the given integer.

23�460,04623 ƒ 461019�3,800,0187 ƒ 280021

85_ _1

n

RemainderWhen n Is

Divided by 9Sum of theDigits of n

Remainder When the

Sum of the Digits of n IsDivided by 9

a. 31b. 143c. 345d. 2987e. 7652

f. Make a conjecture about the remainder and the sum ofthe digits in an integer when they are divided by 9.

13. A palindrome is a number that reads the same forward asbackward.a. Check the following four-digit palindromes for divis-

ibility by 11:i. 4554 ii. 9339 iii. 2002 iv. 2222

b. Are all four-digit palindromes divisible by 11? Whyor why not?

c. Are all five-digit palindromes divisible by 11? Whyor why not?

d. Are all six-digit palindromes divisible by 11? Why orwhy not?

14. The numbers 5872 and 2785 are a palindromic pairof numbers because reversing the order of the digits ofone number gives the other number. Explain why in a

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palindromic pair, if one number is divisible by 3, then sois the other.

15. Classify each of the following as true or false, assumingthat a, b, c, and d are integers. If a statement is false, givea counterexample.a. If and , then for any integers x

and y.b. If and then c. If then .d. If then

16. Prove Theorem 5–13(b).17. a. Choose a two-digit number such that the number in

the tens place is 1 greater than the number in the unitsplace. Reverse the digits in your number, and subtractthis number from your original number; for example,

d�a2.d�a,d ƒ ad ƒ a2,

d�1a + b2.d�b,d�a

d ƒ 1ax + by2d ƒ bd ƒ a

Make a conjecture concerning theresult of performing this operation.

b. Choose any two-digit number such that the number inthe tens place is 2 greater than the number in the unitsplace. Reverse the digits in your number, and subtractthis number from your original number; for example,

. Make a conjecture concerning the re-sult of performing this operation.

c. Prove that for any two-digit number, if the digits arereversed and the numbers subtracted, the differenceis a multiple of 9.

d. Investigate what happens whenever two-digit num-bers with equal digit sums are subtracted; for example,62 - 35 = 27.

31 - 13 = 18

87 - 78 = 9.


Communication1. A customer wants to mail a package. The postal clerk

determines the cost of the package to be $18.95, butonly 6¢ and 9¢ stamps are available. Can the availablestamps be used for the exact amount of postage for thepackage? Why or why not?

2. a. Jim uses his calculator to see if a number n having eightor fewer digits is divisible by a number d. He finds that

has a display of 32. Does ? Why?b. If gives a display of 16.8, does ? Why?

3. Is the area of each of the following rectangles divisibleby 4? Explain why or why not.a.

d ƒ nn , dd ƒ nn , d

414,143,313 are divisible by 3. Explain why this state-ment is true.

7. Enter any three-digit number on the calculator; for exam-ple, enter 243. Repeat it: 243,243. Divide by 7. Divide by11. Divide by 13. What is the answer? Try it again withany other three-digit number. Will this always work?Why?

8. Alexa claims that she can justify the divisibility test by11. She says: I noticed that each even power of 10 can bewritten as a multiple of 11 plus 1 and every odd power of 10can be written as a multiple of 11 minus 1. In fact:

and so on.Now I look at a four-digit number abcd and proceed as in the

divisibility by 3. I collect the parts that are divisible by 11 regard-less of what the digits are and put together the rest, which is

Complete the details of Alexa’s argument and justify thetest for divisibility by 11.

9. Take a number written in base ten with three or moredigits and subtract the units digit from the indicatedexpression. By what numbers can you be sure that thedifference is divisible? Justify your answers.a. The units digitb. The number formed by the last two digits (that is, the

tens digit followed by the units digit).

d - c + b - a

= 900 # 11 + 9 # 11 + 1 = 909 # 11 + 1 104 = 102 # 102 = 10019 # 11 + 12

= 90 # 11 + 11 - 1 = 91 # 11 - 1 103 = 10 # 102 = 1019 # 11 + 12 = 90 # 11 + 10 102 = 99 + 1 = 9 # 11 + 1 10 = 11 - 1

52,832 cm

324,518 cm

52,834 cm

324,514 cm

b.

4. Can you find three consecutive natural numbers none ofwhich is divisible by 3? Explain your answer.

5. Answer each of the following and justify your answers.a. If a number is not divisible by 5, can it be divisible by

10?b. If a number is not divisible by 10, can it be divisible

by 5?6. A number in which each digit except 0 appears exactly

3 times is divisible by 3. For example, 777,555,222 and

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c. The sum of the digitsd. Answer the preceding questions for a three- or more

digit number written in base five.10. a. In what bases will divisibility by 2 depend only on the

last digit? Justify your answer.b. In what bases will divisibility by 2 depend only on the

sum of the digits being even or odd? Justify youranswer.

Open-Ended11. A breakfast-food company had a contest in which

numbers were placed in breakfast-food boxes. A prizeof $1000 was awarded to anyone who could collectnumbers whose sum was 100. The company had thou-sands of cards made with the following numbers onthem:

a. If the company did not make any more cards, is therea winning combination?

b. If the company is going to add one more number tothe list and it wants to make sure the contest has atmost 1000 winners, suggest a strategy for it to use.

12. How would you use concrete materials to explain toyoung children the following:a. A number being even or oddb. A number being divisible by 3 or not being divisible

by 3c. That if then

Cooperative Learning13. In your group, discuss the value of teaching various

divisibility tests in middle school. If a teacher decidesto discuss the various tests, how should they beintroduced?

Questions from the Classroom14. Jane claimed that a number is divisible by 4 if each of the

last two digits is divisible by 4. Is this claim accurate? Ifnot, how would you suggest that Jane change it to makeit accurate?

15. Jim says that and mean the same thing. Howwould you respond?

16. Betty noticed that , , and . She noticedthat . She then noticed that or soshe thought or 24 should divide 36. What do youtell her?

17. A student claims that and implies , andhence, . Is the student correct?

18. A student writes, “If and , then .”How do you respond?

19. Your seventh-grade class has just completed a unit on di-visibility rules. One of the better students asks why di-visibility by numbers other than 3 and 9 cannot be testedby dividing the sum of the digits by the tested number.How should you respond?

d�1a + b2d� bd�aa ƒ 0

a ƒ 1a - a2a ƒ aa ƒ a

4 # 66 ƒ 364 ƒ 3618 = 2 # 9

18 ƒ 369 ƒ 362 ƒ 36

a>ba ƒ b

2 ƒ a4 ƒ a,

3 12 15 18 27 33 45 51 66 75 84 90

20. A student says that a number with an even number ofdigits is divisible by 7 if, and only if, each of the numbersformed by pairing the digits into groups of two is divisi-ble by 7. For example, 49,562,107 is divisible by 7, sinceeach of the numbers 49, 56, 21, and 07 is divisible by 7.Is this true?

21. A student claims that a number is divisible by 21 if, andonly if, it is divisible by 3 and by 7, and, in general, anumber is divisible by if, and only if, it is divisible bya and by b. What is your response?

22. A student found that all three-digit numbers of theform aba, where is a multiple of 7, are divisibleby 7. She would like to know why this is so. How do yourespond?

Review Problems23. Find all integers x (if possible) that make each of the

following true:a.b.c.d.e.f.

24. Simplify each of the following:a.b.c.d.

25. Consider the function whose do-main is the set of integers, and answer the followingquestions:a. Find b. For what value of x will the value of y be 17?c. Is 2 a possible output? Explain your answer.d. Find the range of the function.


Write each of the following numbers in the circle whereit belongs.

f 1-52.

y = f 1x2 = -2x - 31x - 122 - x2 + 2xy - x - 21y - x21-2x22 - 3x23x - 11 - 2x2

-x 6 0

- ƒ -x ƒ = 5

-1x - 12 = 1 - x1-x2 , 0 = -11-22 ƒ x ƒ = 631-x2 = 6

a + b

a # b

30, 47, 124

27

53 642

38

1

Odd Numbers Even Numbers

NAEP, 2007, Grade 4

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5-4 Prime and Composite Numbers

Principles and Standards states:

Students should recognize that different types of numbers have particular characteristics; forexample, square numbers have an odd number of factors and prime numbers have only twofactors.

One method used in elementary schools to determine the positive factors of a natural numberis to use squares of paper or cubes and to represent the number as a rectangle. Such a rectangleresembles a candy bar formed with small squares. The dimensions of the rectangle are divisorsor factors of the number. For example, Figure 5-20 shows rectangles to represent 12.

1p. 1512

BRAIN TEASER Dee finds that she has an extraordinary Social Security number. Its nine digits contain all the numbers from 1 through 9. They also form a number with the following characteristics:When read from left to right, its first two digits form a number divisible by 2, its first three digits forma number divisible by 3, its first four digits form a number divisible by 4, and so on, until the completenumber is divisible by 9. What is Dee’s Social Security number?

1

12

6

2

4

3

Figure 5-20

As the figure shows, the number 12 has six positive divisors: 1, 2, 3, 4, 6, and 12. If rectan-gles were used to find the divisors of 7, then we would find only a rectangle, asFigure 5-21 shows. Thus, 7 has exactly two divisors: 1 and 7.

1 * 7

To illustrate further the number of positive divisors of a natural number, we constructTable 5-1. Below each number listed across the top, we identify numbers less than or equalto 37 that have that number of positive divisors. For example, 12 is in the 6 column be-cause it has six positive divisors, and 7 is in the 2 column because it has only two positivedivisors.

1

7

Figure 5-21

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Section 5-4 Prime and Composite Numbers 301

1 2 3 4 5 6 7 8 9

1 2 4 6 16 12 24 363 9 8 18 305 25 10 207 14 28

11 15 3213 2117 2219 2623 2729 3331 3437 35

Table 5-1 Number of Positive Divisors

The numbers in the 2 column in Table 5-1 are of particular importance. Notice that theyhave exactly two positive divisors, namely, 1 and themselves. Any positive integer withexactly two distinct, positive divisors is a prime number, or a prime. Any integer greater than1 that has a positive factor other than 1 and itself is a composite number, or a composite. Forexample, 4, 6, and 16 are composites because they have positive factors other than 1 andthemselves. The number 1 has only one positive factor, so it is neither prime nor compos-ite. From the 2 column in Table 5-1, we see that the first 12 primes are 2, 3, 5, 7, 11, 13, 17,19, 23, 29, 31, and 37. The number 2 is sometimes called the “oddest” prime because it isthe only even prime. Other patterns in the table are explored in the problem set.

Show that the following numbers are composite:

a. 1564b. 2781c. 1001d.

Solution a. Since 1564 is divisible by 2 and is composite.b. Since 2781 is divisible by 3 and is composite.c. Since 1001 is divisible by 11 and is composite.d. Because a product of odd numbers is odd (why?), is odd. If we

add 1 to an odd number, the sum is even. An even number (other than 2) hasa factor of 2 and is therefore composite.

3 # 5 # 7 # 11 # 1311 ƒ 311 + 02 - 10 + 124,3 ƒ 12 + 7 + 8 + 12,2 ƒ 4,

3 # 5 # 7 # 11 # 13 + 1

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-24▲

▲

NOW TRY THIS 5-11

a. What patterns do you see forming in Table 5-1?b. Will there be other entries in the 1 column? Why?c. What are the next three numbers in the 3 column?d. Find an entry for the 7 column.e. What kinds of numbers have an odd number of factors? Why?

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NOW TRY THIS 5-12 The FoxTrot cartoon concerns divisibility and prime numbers. Answer thefollowing questions based on the cartoon.

a. Select a number at random from the cartoon and divide it by 13; then divide it by 17; then divide it by 19.Keep doing this until you find a number that when divided by 13, 17, or 19 gives you a whole-numberanswer. What does it mean when you get a whole-number answer?

b. The cartoonist made one mistake. The number 2261 appears in the left center of the cartoon. Explainwhy this number can’t be included in the cartoon.

Prime FactorizationIn the grade 7 Focal Points we find this statement:

Students continue to develop their understanding of multiplication and division and the struc-ture of numbers by determining if a counting number greater than I is a prime, and if it is not, byfactoring it into a product of primes. (p. 19)

Composite numbers can be expressed as products of two or more whole numbers greaterthan 1. For example, or Each expression of 18 as aproduct of factors is a factorization.

A factorization containing only prime numbers is a prime factorization. To find a primefactorization of a given composite number, first rewrite the number as a product of twosmaller numbers greater than 1. Continue the process, factoring the lesser numbers untilall factors are primes. For example, consider 260:

The procedure for finding a prime factorization of a number can be organized using a factortree, as Figure 5-22(a) demonstrates. The last branches of the tree display the prime factorsof 260. A second way to factor 260 is shown in Figure 5-22(b). The two trees produce thesame prime factorization, except for the order in which the primes appear in the products.

260 = 26 # 10 = 12 # 13212 # 52 = 2 # 2 # 5 # 13 = 22 # 5 # 13

18 = 2 # 3 # 3.18 = 2 # 9, 18 = 3 # 6,

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The Fundamental Theorem of Arithmetic assures us that once we find a prime factoriza-tion of a number, a different prime factorization of the same number cannot be found. Forexample, consider 260. We start with the least prime, 2, and see whether it divides 260. Ifnot, we try the next greater prime and check for divisibility by this prime. Once we find aprime that divides the number in question, we must find the quotient of the number di-vided by the prime. This step in the prime factorization of 260 is shown in Figure 5-23(a).Next we check whether the prime divides the quotient. If so, we repeat the process; if not,we try the next greater prime, 3, and check to see if it divides the quotient. We see that 130divided by 2 yields 65, as shown in Figure 5-23(b). We continue the procedure, usinggreater primes, until a quotient of 1 is reached. The original number is the product of allthe prime divisors used. The complete procedure for 260 is shown in Figure 5-23(c). Analternative form of this procedure is shown in Figure 5-23(d).

The primes in the prime factorization of a number are typically listed in increasing orderfrom left to right, and if a prime appears in a product more than once, exponential notation isused. Thus, the factorization of 260 is written as Prime factorization is demonstratedin the student page on page 304. Notice that the factor tree is developed in two different waysleading to the same result. Work through the divided practice on the student page.

22 # 5 # 13.

Theorem 5–23Fundamental Theorem of Arithmetic Each composite number can be written as a productof primes in one, and only one, way except for the order of the prime factors in the product.

260

26 10

132 2 5

(a)

260

5 52

2 26

(b)2 13

Figure 5-22

2 260

130

(a)

2 260

130

(b)

2

65

2 260

65

(c)

2

5

130

1313

1

260

130

65

13

2

2

5

113

(d) Alternative form

Figure 5-23

The Fundamental Theorem of Arithmetic, or the Unique Factorization Theorem, states thatin general, if order is disregarded, the prime factorization of a number is unique.

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Source: Scott Foresman-Addison Wesley, enVisionMATH, 2008, Grade 6 (p. 124).

School Book Page HOW CAN YOU WRITE A NUMBERAS A PRODUCT OF PRIME FACTORS?

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The positive divisors of 24 occur in pairs, where the product of the divisors in each pair is24. If 3 is a divisor of 24, then , or 8, is also a divisor of 24. In general, if a natural num-ber k is a divisor of 24, then is also a divisor of 24.

Another way to think of the number of positive divisors of 24 is to consider the prime fac-torization The positive divisors of are and The positive divisorsof 3 are and We know that has or 4, divisors and has or 2, di-visors. Because each divisor of 24 is the product of a divisor of and a divisor of then weuse the Fundamental Counting Principle to conclude that 24 has or 8, positive divisors.This is summarized in Table 5-2.

4 # 2,31,23

11 + 12,3113 + 12,2331.3023.20, 21, 22,2324 = 23 # 3.

24>k24>3

NOW TRY THIS 5-13 Colored rods are used in the elementary-school classroom to teach many con-cepts. The rods vary in length from 1 cm to 10 cm. Various lengths have different colors; for example, the 5 rod is yellow. The rods and their colors are shown in Figure 5-24. A row with all the same color rods iscalled a one-color train.

a. What rods can be used to form a one-color train for 18?b. What one-color trains are possible for 24?c. How many one-color trains of two or more rods are possible for each prime number?d. If a number can be represented by an all-red train, an all-green train, and an all-yellow train, what is

the least number of factors it must have? What are they?

Ora

nge

Blu

e

Bro

wn

Bla

ck

Dar

k G

reen

Yel

low

Purp

le

Gre

en

Red

Whi

te1 2 3 4 5 6 7 8 9 10

Figure 5-24

Positive Divisors of 23 20 = 1 21 = 2 22 = 4 23 = 8

Positive Divisors of 31 30 = 1 31 = 3

Positive Divisors of Positive Divisors of 23

31 #31 # 20 = 330 # 20 = 1

31 # 21 = 630 # 21 = 2

31 # 22 = 1230 # 22 = 4

31 # 23 = 2430 # 23 = 8

(Positive Divisors of 24)

Table 5-2

Number of DivisorsHow many positive divisors does 24 have? Note that the question asks for the number ofdivisors, not just prime divisors. To aid in the listing, we group divisors as follows:

1, 2, 3, 4, 6, 8, 12, 24

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Determining If a Number Is PrimeAs depicted in the following cartoon by Sidney Harris, prime numbers have fascinated peo-ple of various backgrounds. In Now Try This 5-14, you might have found that to determineif a number is prime, you must check only divisibility by prime numbers less than the givennumber. Why? However, do we need to check all the primes less than the number? Supposewe want to check if 97 is prime and we find that 2, 3, 5, and 7 do not divide 97. Could agreater prime divide 97? If p is a prime greater than 7, then If then alsodivides 97. However, because then must be less than 10 and hence cannotdivide 97. Why? So we see that there is no need to check for divisibility by numbers otherthan 2, 3, 5, and 7. These ideas are generalized in the following theorems.

97>pp Ú 1197>pp ƒ 97,p Ú 11.

Find the number of positive divisors of each of the following:

a. 1,000,000 b.

Solution a. We first find the prime factorization of 1,000,000.

Because has divisors and has divisors, then by the Fun-damental Counting Principle has or 49, divisors.

b. The prime factorization of 210 is

By the Fundamental Counting Principle, the number of divisors of is110 + 12110 + 12110 + 12110 + 12 = 114 = 14,641.

21010

21010 = 12 # 3 # 5 # 7210 = 210 # 310 # 510 # 710 210 = 21 # 10 = 3 # 7 # 2 # 5 = 2 # 3 # 5 # 7,

16 + 1216 + 12,26 # 566 + 1566 + 126

= 26 # 56 = 12 # 2 # 2 # 2 # 2 # 2215 # 5 # 5 # 5 # 5 # 52

1,000,000 = 106 = 12 # 526 = 12 # 5212 # 5212 # 5212 # 5212 # 5212 # 52

21010▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-25

▲

▲

NOW TRY THIS 5-14 To determine whether it is necessary to divide 97 by 2, 3, 4, 5, 6, , 96 to checkif it is prime, answer the following (justify your answers):

a. If 2 is not a divisor of 97, could any multiple of 2 be a divisor of 97?b. If 3 is not a divisor of 97, what other numbers could not be divisors of 97?c. If 5 is not a divisor of 97, what other numbers could not be divisors of 97?d. If 7 is not a divisor of 97, what other numbers could not be divisors of 97?e. Conjecture what numbers we have to check for divisibility in order to determine if 97 is prime.

Á

Theorem 5–24If p and q are different primes, then has positive divisors. In general, if are primes and are whole numbers, then has

positive divisors.1n1 + 121n2 + 12 # Á # 1nk + 12p1

n1 # p2n2 # Á # pk

nkn1, n2 Á , nkp1, p2 Á , pk

1n + 121m + 12pnqm

This discussion can be generalized as follows: If p is any prime and n is any natural num-ber, then the positive divisors of are Therefore, there are positive

divisors of . Now, using the Fundamental Counting Principle, we can find thenumber of divisors of any number whose prime factorization is known.

pn1n + 12p0, p1, p2, p3, Á , pn.pn

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Theorem 5–26 can be used to help determine whether a given number is prime or com-posite. For example, consider the number 109. If 109 is composite, it must have a prime di-visor p such that The primes whose squares do not exceed 109 are 2, 3, 5, and 7.Mentally, we can see that and Hence, 109 is prime. The argu-ment used leads to the following theorem.

7�109.2�109, 3�109, 5�109,p2 … 109.

Theorem 5–26If n is composite, then n has a prime factor p such that p2 … n.

Theorem 5–25If d is a divisor of n, then is also a divisor of n.

nd

Suppose that p is the least divisor of n (greater than 1). Such a divisor must be prime(why?). Then Since and p was the least divisor of Therefore,

Since we have This idea is summarized in the followingtheorem.

p2 … n.n Ú p2,n = pk Ú pp = p2.n, k Ú p.k ƒ nn = pk, k Z 1.

Theorem 5–27If n is an integer greater than 1 and not divisible by any prime p, such that then n isprime.

p2 … n,

R E M A R K Because implies that , Theorem 5–27 says that to determineif a number n is prime, it is enough to check if any prime less than or equal to is adivisor of n.

1np … 1np2 … n

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One way to find all the primes less than a given number is to use the Sieve of Eratosthenes,named after the Greek mathematician Eratosthenes (ca. 276–194 BCE). If all the natural num-bers greater than 1 are considered (or placed in the sieve), the numbers that are not prime aremethodically crossed out (or drop through the holes of the sieve). The remaining numbersare prime. The partial student page on page 309 illustrates this process. Before reading on,build a sieve, work through the process, and answer the questions on the student page.

The Sieve of Eratosthenes is another way to motivate Theorem 5–27. Notice the obser-vations from the sieve in Table 5-3 as we crossed out numbers.

a. Is 397 composite or prime? b. Is 91 composite or prime?

Solution a. The possible primes p such that are 2, 3, 5, 7, 11, 13, 17, and 19. Be-cause and thenumber 397 is prime.

19�397,2�397, 3�397, 5�397, 7�397, 11�397, 13�397, 17�397,p2 … 397

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-26

▲

▲

b. The possible primes p such that are 2, 3, 5, and 7. Because 91 isdivisible by 7, it is composite.

p2 … 91

Prime Observation

2 First number not crossed out that 2 divides is 4 = 22.3 First number not crossed out that 3 divides is 9 = 32.5 First number not crossed out that 5 divides is 25 = 52.7 First number not crossed out that 7 divides is 49 = 72.

Table 5-3

We didn’t need to continue the procedure for 11 because the first number not crossed outthat 11 divides is , or 121, and the table only goes to 100. Therefore, to test whether anumber such as 137 is a prime, we first test for divisibility by all primes up to but not includ-ing the first prime whose square is greater than 137. Because , any prime greaterthan or equal to 13 would give a quotient less than or equal to 13, and we have alreadychecked these primes. This shows that when testing to see if a number is prime, we need tryas divisors only primes whose squares are less than or equal to the number being tested.

More About PrimesThere are infinitely many whole numbers, infinitely many odd whole numbers, and infi-nitely many even whole numbers. Are there infinitely many primes? Because prime numbersdo not appear in any known pattern, the answer to this question is not obvious. Euclid wasthe first to prove that there are infinitely many primes.

132 = 169

112

HistoricalNote

Eratosthenes (276–194 BCE), a Greek scholar, was born in Cyrene but spent most of his lifein Alexandria as the chief librarian at the museum there. In his work Geographica, he gavearguments for the spherical shape of the Earth. Today, Eratosthenes is best known for his“sieve”—a systematic procedure for isolating the prime numbers—and for a simple methodfor calculating the circumference of the Earth. ▲▲

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Mathematicians have long looked for a formula that produces only primes, but no onehas found one. One result was the expression where n is a whole number.Substituting for n in the expression always results in a prime number.However, substituting 41 for n gives or a composite number. In 1998,Roland Clarkson, a 19-year-old student at California State University, showed that

is prime. The number has 909,526 digits. The full decimal expansion of the num-ber would fill several hundred pages. Since then, more large primes have been discovered:

and These are examplesof Mersenne primes. A Mersenne prime, named after the French monk Marin Mersenne

230,402,457 - 1 19,152,052 digits2.232,582,657 - 1 19,808,358 digits2

23021377 - 1

412,412 - 41 + 41,0, 1, 2, 3, Á , 40

n2 - n + 41,

Source: Scott Foresman-Addison Wesley, Mathematics, 2008, Grade 6 ( p. 149).

School Book Page SIEVE OF ERATOSTHENES

▲▲


Sophie Germain (1776–1831) was born in Paris and grew up during the French Revolution.She wanted to study at the prestigious École Polytechnique but women were not allowedas students. Consequently, she studied from lecture notes and from Gauss’s monograph onnumber theory. She made major contributions to the mathematical theory of elasticity, forwhich she was awarded the prize of the French Academy of Sciences. Germain’s work washighly regarded by Gauss, who recommended her for an honorary degree from the University of Göttingen. She died before the degree could be awarded.IS

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(1588–1648), is a prime of the form where n is prime. On August 23, 2008, a UCLAcomputer discovered the 45th known Mersenne prime, On September 6, 2008, the 46th Mersenne prime, wasdiscovered in Germany.

Searching for large primes has led to advances in distributed computing, that is, using theInternet to engage the unused computing power of great numbers of computers. Searchingfor Mersenne primes has been used as a test for computer hardware.

Another type of interesting prime is a Sophie Germain prime, which is an odd prime p forwhich is also a prime. Notice that is a Sophie Germain prime, since or 7, is also a prime. Check that 5, 11, and 23 are also such primes. The primes were namedafter the French mathematician Sophie Germain. In 2007, the greatest Sophie Germainprime discovered had 51,910 digits.

2 # 3 + 1,p = 32p + 1

1 111,185,272 digits2237,156,661 -1 112,978,189 digits2.243,112,609 -

2n - 1,

Problem Solving How Many Bears?

A large toy store carries one kind of stuffed bear. On Monday the store sold a certain num-ber of the stuffed bears for a total of $1843 and on Tuesday, without changing the price, thestore sold a certain number of the stuffed bears for a total of $1957. How many toy bearswere sold each day if the price of each bear is a whole number and greater than $1?

Understanding the Problem One day a store sold a number of stuffed bears for $1843 and onthe next day a number of them for a total of $1957. We need to find the number of bearssold on each day.

Devising a Plan If x bears were sold the first day and y bears the second day, and if the priceof each bear was c dollars, we would have and . Thus, 1843 and 1957should have a common factor—the price c. We could factor each number and find the pos-sible factors. If the problem is to have a unique solution, the two numbers should have onlyone common factor other than 1. Any common factor of 1957 and 1843 will also be a factorof and the factors of 114 are easier to find.

Carrying Out the Plan We have Thus, if 1957 and 1843 have acommon prime factor, it must be 2, 3, or 19. But neither 2 nor 3 divides the numbers, hencethe only possible common factor is 19. We divide each number by 19 and find

1957 = 19 # 103 1843 = 19 # 97

114 = 2 # 57 = 2 # 3 # 19.

1957 - 1843 = 114

cy = 1957cx = 1843


In the 1970s, determining large prime numbers became extremely useful in coding anddecoding secret messages. In all coding and decoding, the letters of an alphabet correspondin some way to nonnegative integers. A “safe” coding system, in which messages are unintel-ligible to everyone except the intended receiver, was devised by three Massachusetts Insti-tute of Technology scientists (Ronald Rivest, Adi Shamir, and Leonard Adleman) and isreferred to as the RSA (their initials) system. The secret deciphering key consists of twolarge prime numbers chosen by the user. The enciphering key is the product of these twoprimes. Because it is extremely difficult and time-consuming to factor large numbers, it waspractically impossible to recover the deciphering key from a known enciphering key. In1982, new methods for factoring large numbers were invented, which resulted in the use ofeven greater primes to prevent the breaking of decoding keys. ▲▲

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Notice that neither 97 nor 103 is divisible by 2, 3, 5, or 7. Hence 97 and 103 are primes(why?) and therefore the only common factor (greater than 1) of 1843 and 1957 is 19. Con-sequently, the price of each bear was $19. The first day 97 bears were sold and the next day103 bears were sold.

Looking Back Notice that the problem had a unique solution because the only common fac-tor (greater than 1) of the two numbers was 19. We could create similar problems by havingthe price of the item be a prime number and the number of items sold each day also be primenumbers. For example, the total sale on the first day could have been , or $2323 andon the second day , or $2461 (notice that 23, 101, and 107 are prime numbers).

To find a common factor of 1957 and 1843, we found all the common factors ofand checked which of the factors of the difference was a

common factor of the original numbers. We have used Theorem 5–13: If and then This theorem assures us that every common factor of a and b will also bea factor of .a - b

d ƒ 1a - b2.d ƒ b,d ƒ a

1957 - 1843 = 114 = 2 # 3 # 19

23 # 10723 # 101

▲▲

Assessment 5-4A

1. Find the least positive number that is divisible by threedifferent primes.

2. Determine which of the following numbers are primes:a. 109 b. 119c. 33 d. 101e. 463 f. 97g. h.

3. Use a factor tree to find the prime factorization for eachof the following:a. 504 b. 2475 c. 11,250

4. a. Fill in the missing numbers in the following factortree:

2 # 3 # 5 # 7 - 12 # 3 # 5 # 7 + 1

7. a. When the U.S. flag had 48 stars, the stars formed arectangular array. In what other rectangular

arrays could they have been arranged?b. How many rectangular arrays of stars could there be

if there were only 47 states?8. a. Use the Fundamental Theorem of Arithmetic to jus-

tify that if 2 n and 3 n, then 6 n.b. Is it always true that if a n and b n, then ab n2? Either

prove the statement or give a counterexample.9. Mr. Arboreta wants to plant fruit trees in a rectangular

array. For each of the following numbers of trees, find allpossible numbers of rows if each row is to have the samenumber of trees:a. 36b. 28c. 17d. 144

10. Some of the divisors of a locker number are 2, 5, and 9.If there are exactly nine additional positive divisors, whatis the locker number?

11. Extend the Sieve of Eratosthenes to find all primes be-tween 100 and 200.

12. The prime numbers 11 and 13 are called twin primesbecause they differ by 2. (The existence of infinitelymany twin primes has not been proved.) Find all thetwin primes less than 200.

13. If what other positive integers divide n?14. If 1000 is a factor of n, what other positive integers di-

vide n? How many such integers are there?15. It is not known whether there are infinitely many primes

in the infinite sequence consisting of the numbers whoseonly digits are ones; 1, 11, 111, 1111, . . . . Find infi-nitely many composite numbers in the sequence.

42 ƒ n,

ƒƒƒƒƒƒ

6 by 8

5

73

2

b. How could you find the top number without findingthe other two numbers?

5. What is the greatest prime you must consider to testwhether 5669 is prime?

6. Find the prime factorizations of the following:a.b.c. 251d. 1001

102 # 26 # 49101 # 2 # 3 # 4 # 5 # 6 # 7 # 8 # 9 # 10

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16. Is a factor of Explain why or why not.17. Explain why each of the following numbers is composite:

a.b.c.d. (Note:

18. Explain why is not a prime factorization andfind the prime factorization of the number.

23 # 32 # 25310! = 10 # 9 # 8 # 7 # 6 # 5 # 4 # 3 # 2 # 1.210! + 7

13 # 5 # 7 # 11 # 132 + 513 # 4 # 5 # 6 # 7 # 82 + 23 # 5 # 7 # 11 # 13

34 # 27?32 # 24 19. Find the prime factorizations of each of the following:a.b.c.d.

20. I am a prime number greater than 40 and less than 90.My units digit and my tens digit are both prime. Thedifference between my units digit and tens digit is not 2.What number am I?

2 # 3 # 5 # 7 # 11 + 12 # 34 # 5110 # 7 + 4 # 34 # 511010060 # 300403610 # 4920 # 615

Assessment 5-4B

1. Determine which of the following numbers are primes:a. 89 b. 147c. 159 d. 187e. f.

2. Use a factor tree to find the prime factorization for eachof the following:a. 304b. 1570c. 9550

3. a. Fill in the missing numbers in the following factor tree:

2 # 3 # 5 # 7 - 52 # 3 # 5 # 7 + 5

7. Find the least number divisible by each natural numberless than or equal to 12.

8. Find the greatest four-digit number that has exactlythree positive factors.

9. Show that if 1 were considered a prime, every numberwould have more than one prime factorization.

10. Is it possible to find positive integers x, y, and z such thatWhy or why not?

11. a. Show that there are infinitely many composite num-bers in the arithmetic sequence

b. Does every arithmetic sequence consisting of inte-gers with difference greater than 0 have infinitelymany composite numbers? Justify your answer.

12. If explain why isa factor of N.

13. Is a factor of Explain why or why not.14. Explain why each of the following numbers is composite:

a.b. where or 10

15. Explain why is not a prime factorization andfind the prime factorization of the number.

16. A prime such as 7331 is a superprime because any inte-gers obtained by deleting digits from the right of 7331are prime; namely, 733, 73, and 7.a. For a prime to be a superprime, what digits cannot

appear in the number?b. Of the digits that can appear in a superprime, what digit

cannot be the leftmost digit of a superprime?c. Find all of the two-digit superprimes.d. Find a three-digit superprime other than 733.

17. Is the following always true? ( Justify your answer.) If, then or .

18. Find the prime factorizations of each of the following:a.b.c. 22 # 35 # 755 + 24 # 34 # 755

84 # 325164 # 814 # 66

m ƒ bm ƒ am ƒ ab

22 # 53 # 92k = 2, 3, 4, 5, 6, 7, 8, 9,10! + k,

7 # 11 # 13 # 17 + 17

33 # 22?32 # 24

2 # 3 # 5 # 7 # 112N = 26 # 35 # 54 # 73 # 117,

1, 5, 9, 13, 17, Á .

2x # 3y = 5z?

3

32

2

b. How could you find the top number without findingthe other two numbers?

4. What is the greatest prime you must consider to testwhether 503 is prime?

5. Find the prime factorizations of the following:a. 1001b.c.d.

6. Suppose the 435 members of the House of Representa-tives are placed on committees consisting of more than2 members but fewer than 30 members. Each committeeis to have an equal number of members and each mem-ber is to be on only one committee.a. What size committees are possible?b. How many committees are there of each size?

11110 - 11199991010012

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Communication1. Explain why the product of any three consecutive integers

is divisible by 6.2. Explain why the product of any four consecutive integers

is divisible by 24.3. In order to test for divisibility by 12, one student checked

to determine divisibility by 3 and 4; another checked fordivisibility by 2 and 6. Are both students using a correctapproach to divisibility by 12? Why or why not?

4. In the Sieve of Eratosthenes for numbers less than 100,explain why, after we cross out all the multiples of 2, 3, 5,and 7, the remaining numbers are primes.

5. Let Without multi-plying, show that none of the primes less than or equal to19 divides M.

6. A woman with a basket of eggs finds that if she removesthe eggs from the basket 3 or 5 at a time, there is always 1egg left. However, if she removes the eggs 7 at a time,there are no eggs left. If the basket holds up to 100 eggs,how many eggs does she have? Explain your reasoning.

7. Explain why, when a number is composite, its least posi-tive divisor, other than 1, must be prime.

8. Euclid proved that given any finite list of primes, there ex-ists a prime not in the list. Read the following argumentand answer the questions that follow.

be a list of all the primes less thanor equal to a certain prime p. We will show that there ex-ists a prime not on the list. Consider the product

Notice that every prime in our list divides that product.However, if we add 1 to the product, that is, form thenumber then none of theprimes in the list will divide N. Notice that whether N isprime or composite, some prime q must divide N. Becauseno prime in our list divides N, q is not one of the primesin our list. Consequently . We have shown thatthere exists a prime greater than p.a. Explain why no prime in the list will divide N.b. Explain why some prime must divide N.c. Someone discovered a prime that has 65,050 digits.

How does the preceding argument assure us thatthere exists an even larger prime?

q 7 p

N = 12 # 3 # 5 # 7 # Á # p2 + 1,

2 # 3 # 5 # 7 # Á # p

Let 2, 3, 5, 7, Á , p

M = 2 # 3 # 5 # 7 + 11 # 13 # 17 # 19.

d. Does the argument show that there are infinitelymany primes? Why or why not?

e. Let Without mul-tiplying, explain why some prime greater than 19 willdivide M.

Open-Ended9. a. In which of the following intervals do you think there

are the most primes? Why? Check to see if you werecorrect.

i. 0–99 ii. 100–199b. What is the longest string of consecutive composite

numbers in the intervals?c. How many twin primes are there in each interval?d. What patterns, if any, do you see for any of the pre-

ceding questions? Predict what might happen inother intervals.

10. A number is a perfect number if the sum of its factors(other than the number itself) is equal to the number. Forexample, 6 is a perfect number because its factors sum to6, that is, An abundant number has fac-tors whose sum is greater than the number itself. Adeficient number is a number with factors whose sum isless than the number itself.a. Classify each of the following numbers as perfect,

abundant, or deficient:i. 12 ii. 28 iii. 35

b. Find at least one more number that is deficient andone that is abundant.

Cooperative Learning11. A class of 23 students was using square tiles to build rec-

tangular shapes. Each student had more than 1 tile andeach had a different number of tiles. Each student wasable to build only one shape of rectangle. All tiles had tobe used to build a rectangle and the rectangle could nothave holes. For example, a 2 by 6 rectangle uses 12 tilesand is considered the same as a 6 by 2 rectangle but isdifferent from a 3 by 4 rectangle. The class did the activ-ity using the least number of tiles. How many tiles didthe class use? Explore the various rectangles that couldbe made.

1 + 2 + 3 = 6.

M = 2 # 3 # 5 # 7 # 11 # 13 # 17 # 19 + 1.

19. Use the Fundamental Theorem of Arithmetic to justifythe following statement for whole numbers a and bgreater than 1: If p is prime and , then or .p ƒ bp ƒ ap ƒ ab

20. The product of three prime numbers that are less than30 is 1955. What are the three primes?

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Questions from the Classroom12. Mary says that her factor tree for 72 begins with 3 and

24 so her prime factors will be different from Larry’s be-cause he is going to start with 8 and 9. What do you tellMary?

13. Bob says that to check if a number is prime he just usesthe divisibility rules he knows for 2, 3, 4, 5, 6, 8, and 10.He says if the number is not divisible by these numbers,then it is prime. How do you respond?

14. Joe says that every odd number greater than 3 can bewritten as the sum of two primes. To convince the class,he wrote , , and . Howdo you respond?

15. An eighth grader at the Roosevelt Middle School claimsthat because there are as many even numbers as oddnumbers between 1 and 1000, there must be as manynumbers that have an even number of positive divisors asnumbers that have an odd number of positive divisorsbetween 1 and 1000. Is the student correct? Why or whynot?

9 = 7 + 25 = 2 + 37 = 2 + 5

16. A sixth-grade student argues that there are infinitelymany primes because “there is no end to numbers.”How do you respond?

17. A student claims that every prime greater than 3 is aterm in the arithmetic sequence whose nth term is

or in the arithmetic sequence whose nth term is. Is this true? If so why?

Review Problems18. Classify the following as true or false:

a. 11 is a factor of 189.b. 1001 is a multiple of 13.c. and imply

19. Check each of the following for divisibility by 2, 3, 4, 5,6, 7, 8, 9, 10, and 11:a. 438,162b. 2,345,678,910

20. Prove that if a number is divisible by 12, then it is divisi-ble by 3.

21. Could $3376 be divided exactly among either seven oreight people?

7�11001 - 122.7�127 ƒ 1001

6n - 16n + 1

LABORATORY ACTIVITY In Figure 5-25, a spiral starts with 41 at its center and continues in acounterclockwise direction. Primes are shaded. Check the primes along the shaded diagonal. Can you find each of the primes from the formula by substituting appropriate values for n?n2 + n + 41

209210

211

212

213

214

161162

163

164

165

166

121122

123

124

125

126

204205206207208

158159160

43 44

4142

185

186

187

188

189

190

191

192

193

194

195

196

197

238

239

240

241

242

243

244

245

246

247

248

249

250198199200201202203

264265 259260261262263 252 251253254255256257258

141

142

143

144

145

146

105

106

107

108

109

110

111

112

113114115116117118119120

77

78

79

80

81

82

57

58

59

60

616263646566

45

46

47484950

51

52

53 54 55 56

219

220

221

222

223 224 225 226 227 228 229 230 231 232 233 234 235 236 237

169

170

171

172

173 174 175 176 177 178

127

128

93

94

95

96

97 98

67

68

69

70

71 72129

130

131 132 133 134 135 136 137 138

181 182 183 184179 180

101 102 103 10499 100

73 74 75 76

8384858687888990

91

92 147

148

149

150

151152153154155156157

215

216

217

218

167

168

139 140

Figure 5-25

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Section 5-5 Greatest Common Divisor and Least Common Multiple 315

BRAIN TEASER One Saturday Jody cut short her visit with her friend Natasha to take three otherfriends to a movie. “How old are they?” asked Natasha. “The product of their ages is 2450 and the sum is exactly twice your age,” replied Jody. Natasha said: “I need more information.” To that Jody replied, “Ishould have mentioned that I am at least one year younger than the oldest of my three friends.” Withthis information Natasha found the ages of the friends. How did Natasha figure the ages of the friendsand what were their ages?

5-5 Greatest Common Divisor and Least Common Multiple

Consider the following situation:

Two bands are to be combined to march in a parade. A 24-member band will march behind a 30-member band. The combined bands must have the same number of columns. Each column mustbe the same size. What is the greatest number of columns in which they can march?

The bands could each march in 2 columns, and we would have the same number ofcolumns, but this does not satisfy the condition of having the greatest number of columns.The number of columns must divide both 24 and 30. Why? Numbers that divide both 24and 30 are 1, 2, 3, and 6. The greatest of these numbers is 6, so the bands should eachmarch in columns. The first band would have 6 columns with 4 members in each column,and the second band would have 6 columns with 5 members in each column. In this prob-lem, we have found the greatest number that divides both 24 and 30, that is, the greatestcommon divisor (GCD) of 24 and 30.

DefinitionThe greatest common divisor (GCD) of two natural numbers a and b is the greatest naturalnumber that divides both a and b.

In the Research Note we see that greatest common divisors (GCD) and least commonmultiples (LCM) are difficult for students. We provide many methods for finding GCDsand LCMs to help clarify these concepts.

Colored Rods Method

We can build a model of two or more integers with colored rods to determine the GCD oftwo positive integers. For example, consider finding the GCD of 6 and 8 using the 6 rodand the 8 rod, as in Figure 5-26.

Research NotePossibly because stu-dents often confusefactors and multiples,the greatest commonfactor and the leastcommon multipleare difficult topicsfor students to grasp(Graviss and Greaver 1992). ▲▲

▲▲

6 rod 8 rod

(a)

6 rod

(b)

2 rods

8 rod

2 rods

Figure 5-26

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The Intersection-of-Sets Method

In the intersection-of-sets method, we list all members of the set of positive divisors of thetwo integers, then find the set of all common divisors, and, finally, pick the greatest element inthat set. For example, to find the GCD of 20 and 32, denote the sets of divisors of 20 and32 by and respectively.

The set of all common positive divisors of 20 and 32 is

Because the greatest number in the set of common positive divisors is 4, the GCD of 20and 32 is 4, written GCD120, 322 = 4.

D20 ¨ D32 = 51, 2, 46

D32 = 51, 2, 4, 8, 16, 326

D20 = 51, 2, 4, 5, 10, 206

D32,D20

NOW TRY THIS 5-15 Explain how you could use colored rods to solve the marching bands’ problem,stated at the beginning of this section.

NOW TRY THIS 5-16 The Venn diagram in Figure 5-27 shows the factors of 24 and 40. Answer the following:

a. What is the meaning of each of the shaded regions?b. Which factor is the GCD?c. Draw a similar Venn diagram to find the GCD of 36 and 44.

36

1224

1248

Factors of 24 Factors of 40

5102040

Figure 5-27

The Prime Factorization Method

The intersection-of-sets method is rather time-consuming and tedious if the numbers havemany divisors. Another, more efficient, method is the prime factorization method. To findGCD(180, 168), first notice that

and168 = 2 # 2 # 2 # 3 # 7 = 122 # 322 # 7

180 = 2 # 2 # 3 # 3 # 5 = 122 # 323 # 5

To find the GCD of 6 and 8, we must find the longest rod such that we can use multiplesof that rod to build both the 6 rod and the 8 rod. The 2 rods can be used to build both the 6and 8 rods, as shown in Figure 5-26(b); the 3 rods can be used to build the 6 rod but not the8 rod; the 4 rods can be used to build the 8 rod but not the 6 rod; the 5 rods can be used tobuild neither; and the 6 rods cannot be used to build the 8 rod. Therefore, GCD16, 82 = 2.

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We see that 180 and 168 have two factors of 2 and one of 3 in common. These commonprimes divide both 180 and 168. In fact, the only numbers other than 1 that divide both180 and 168 must have no more than two 2s and one 3 and no other prime factors intheir prime factorizations. The possible common divisors are and Hence, the greatest common divisor of 180 and 168 is The procedure for findingthe GCD of two or more numbers by using the prime factorization method is summa-rized as follows:

To find the GCD of two or more positive integers, first find the prime factorizations of the givennumbers and then identify each common prime factor of the given numbers. The GCD is theproduct of the common factors, each raised to the lowest power of that prime that occurs in anyof the prime factorizations.

If we apply the prime factorization technique to finding GCD(4, 9), we see that 4 and 9have no common prime factors. But that does not mean there is no GCD. We still have 1as a common divisor, so Numbers, such as 4 and 9, whose GCD is 1 arerelatively prime. Both the intersection-of-sets method and the prime factorization methodare found on the student page on page 318. Study the page and work through the TalkAbout It questions on the bottom of the student page. Notice that the GCF in the studentpage stands for Greatest Common Factor, which is the same as GCD.

GCD14, 92 = 1.

22 # 3.22 # 3.1, 2, 22, 3, 2 # 3,

Find each of the following:

a.b.c. if and d. if using x and y from e. where and

Solution a. Since and it follows that .

b. Because and it follows that c.d. Because and then

Notice that can also be obtainedby finding the GCD of z and 1274, the answer from .

e. Because x and y have no common prime factors, GCD1x, y2 = 1.1c2

GCD1x, y, z22 # 7 = 14.GCD1x, y, z2 =z = 22 # 7,y = 2 # 73 # 13 # 17,x = 23 # 72 # 11 # 13,

GCD1x, y2 = 2 # 72 # 13 = 1274.GCD10, 132 = 13.13 ƒ 13,13 ƒ 0

22 # 32 = 36GCD1108, 722 =72 = 23 # 32,108 = 22 # 33

y = 310 # 1120x = 54 # 1310GCD1x, y2,1c2z = 22 # 7,GCD1x, y, z2

y = 2 # 73 # 13 # 17x = 23 # 72 # 11 # 13GCD1x, y2GCD10, 132GCD1108, 722▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-27

▲

▲

Calculator Method

Calculators with a key can be used to find the GCD of two numbers. For example, to

find the use the following sequence of buttons to start: First, press

to obtain the display . By pressing the

button, we see on the display as a common divisor of 120 and 180. By pressing the

button again and pressing, , we see 2 again as a factor. The process is re-

peated to reveal 3 and 5 as other common factors. The GCD of 120 and 180 is the productof the common prime factors or 60.2 # 2 # 3 # 5,

x y=Simp

x y2

x yN/D: n/d 60/90=Simp081>0

21GCD1120, 1802,

Simp

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Source: Scott Foresman-Addison Wesley Mathematics 2008, Grade 6 (p. 150).

School Book Page GREATEST COMMON FACTOR

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Some calculators have a GCD feature built in, which you will probably have to go to theMATH menu to find. With this feature, you select GCD and enter the numbers separatedby a comma and closed within a parenthesis; for example, GCD(120, 180). When the ispressed, the GCD of 60 will be displayed.

Euclidean Algorithm Method

Large numbers may be hard to factor. For these numbers, another method is more effi-cient than factorization for finding the GCD. For example, suppose we want to findGCD(676, 221). If we could find two smaller numbers whose GCD is the same asGCD(676, 221), our task would be easier. From Theorem 5–13(c), every divisor of 676 and221 is also a divisor of and 221. Conversely, every divisor of and 221is also a divisor of 676 and 221. Thus, the set of all the common divisors of 676 and 221 is the same as the set of all common divisors of and 221. Consequently,

This process can be continued to subtract three221s from 676 so that Todetermine how many 221s can be subtracted from 676, we could have divided as follows:

When 0 is reached as a remainder, the divisions are complete. Because Based on this illustration, we make the generalization outlined in

the following theorem.GCD1676, 2212 = 13.

GCD10, 132 = 13,

17 R 013�221

3 R 13221�676

GCD1676, 2212 = GCD1676 - 3 # 221, 2212 = GCD113, 2212.GCD1676 - 221, 2212.GCD1676, 2212 =

676 - 221

676 - 221676 - 221

=


Theorem 5–28If a and b are any whole numbers greater than 0 and then where r is the remainder when a is divided by b.

GCD1a, b2 = GCD1r, b2,a Ú b,

R E M A R K Because for all whole numbers x and y not both 0,Theorem 5–28 can be written

GCD1a, b2 = GCD1b, r2.

GCD1x, y2 = GCD1 y, x2

Finding the GCD of two numbers by repeatedly using Theorem 5–28 until the remain-der 0 is reached is referred to as the Euclidean algorithm. This method is found in BookIV of Euclid’s Elements (300 BC). A flowchart for using the Euclidean algorithm is given inFigure 5-28.

Divide thelarger numberby the smaller.

Divide lastdivisor byremainder.

Is theremainder

zero?

No

Yes Last divisoris the GCDof a and b.

Positivenumbers

a and b, a b.

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Use the Euclidean algorithm to find GCD(10764, 2300).

SolutionThus,

Thus,

Thus,

Thus,

Because it follows that GCD110764, 23002 = 92.GCD192, 02 = 92,

GCD1736, 922 = GCD192, 02.8

92�736736 0

GCD11564, 7362 = GCD1736, 922.2

736�15641472

92

GCD12300, 15642 = GCD11564, 7362.1

1564�23001564

736

GCD110764, 23002 = GCD12300, 15642.4

2300�1076492001564

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-28

▲

▲

R E M A R K The procedure for finding the GCD by using the Euclidean algorithm canbe stopped at any step at which the GCD is obvious.

A calculator with the integer division feature can also be used to perform the Euclideanalgorithm. This feature yields the quotient and the remainder when doing a division. Forexample, if the integer division key looks like , then to find weproceed as follows:

which displays 4 1564

which displays 1 736

which displays 2 92

which displays 8 0

The last number we divided by when we obtained the 0 remainder is 92, so

Sometimes shortcuts can be used to find the GCD of two or more numbers, as in thefollowing example.

GCD110764, 23002 = 92

=29INT ,637

=637INT ,4651

=4651INT ,0032

=0032INT ,46701

GCD110764, 23002INT ,

Q

Q

Q

Q

R

R

R

R

Find each of the following:

a.b. The GCD of any two consecutive whole numbers.

Solution a. Any common divisor of three numbers is also a common divisor of any two ofthem (why?). Consequently, the GCD of three numbers cannot be greater

GCD1134791, 6341, 63392▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-29

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than the GCD of any two of the numbers. The numbers 6341 and 6339 areclose to each other and therefore it is easy to find their GCD:

Because cannot be greater than 1, it follows thatit must equal 1.

b. Notice that andIt seems that the GCD of any two consecutive whole

numbers is 1. To justify this conjecture, we need to show that for all wholenumbers n, We have

Least Common MultipleHot dogs are usually sold 10 to a package, while hot dog buns are usually sold 8 to a pack-age. This mismatch causes troubles when one is trying to match hot dogs and buns. Whatis the least number of packages of each you could order so that there is an equal number ofhot dogs and buns? The numbers of hot dogs that we could have are just the multiples of10, that is, Likewise, the possible numbers of buns are 8, 16, 24, 32,

We can see that the number of hot dogs matches the number of buns whenever10 and 8 have multiples in common. This occurs at In this problem, we areinterested in the least of these multiples, 40. Therefore, we could obtain the same numberof hot dogs and buns in the least amount by buying four packages of hot dogs and five pack-ages of buns. The answer 40 is the least common multiple (LCM) of 8 and 10.

40, 80, 120, Á .40, 48, Á .

10, 20, 30, 40, 50, Á .

= 1 = GCD11, n2

GCD1n, n + 12 = GCD1n + 1, n2 = GCD1n + 1 - n, n2

GCD1n, n + 12 = 1.

GCD199, 1002 = 1.GCD16, 72 = 1,GCD15, 62 = 1,GCD14, 52 = 1,

GCD1134791, 6341, 63392

= 1 = GCD12, 63392

GCD16341, 63392 = GCD16341 - 6339, 63392

▲

▲

DefinitionSuppose that a and b are natural numbers. Then the least common multiple (LCM) of a and b isthe least natural number that is simultaneously a multiple of a and a multiple of b.

As with GCDs, there are several methods for finding least common multiples.

Number-Line Method

A number line can be used to find the LCM of two numbers. For example, to find LCM(3, 4),we can show the multiples of 3 and 4 on the number line using intervals of 3 and 4, as shownin Figure 5-29.

0 1 2 3

3 3 3 3 3

4444

4 5 6 7 8 9 10 11 12 13 14 15 16

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Beginning at 0, we see that the arrows do not coincide until the point 12 on the numberline. If the line were continued, the arrows would coincide again at 24, 36, 48, and so on.We see that there are an infinite number of common multiples of 3 and 4, but the leastcommon multiple is 12. Note that this number-line approach is instructive and promotesunderstanding but is not practical for large numbers.

Colored Rods Method

We can use colored rods to determine the LCM of two numbers. For example, consider the3 rod and the 4 rod in Figure 5-30(a). We build trains of 3 rods and 4 rods until they arethe same length, as shown in Figure 5-30(b). The LCM is the common length of the train.

3 rod 4 rod

(a)

10 rod

Length of 12

2 rod

(b)

Three 4 rodsFour 3 rods

Figure 5-30

The Intersection-of-Sets Method

In the intersection-of-sets method, we first find the set of all positive multiples of both the firstand second numbers, then find the set of all common multiples of both numbers, and finallypick the least element in that set. For example, to find the LCM of 8 and 12, denote the setsof positive multiples of 8 and 12 by and respectively.

The set of common multiples is

Because the least number in is 24, the LCM of 8 and 12 is 24, writtenLCM18, 122 = 24.

M8 ¨ M12

M8 ¨ M12 = 524, 48, 72, Á 6

M12 = 512, 24, 36, 48, 60, 72, 84, 96, 108, Á 6

M8 = 58, 16, 24, 32, 40, 48, 56, 64, 72, Á 6

M12,M8

NOW TRY THIS 5-17 Draw a Venn diagram showing and and show how to find LCM(8, 12) using the diagram.

M12M8

The Prime Factorization Method

The intersection-of-sets method for finding the LCM is often lengthy, especially when it isused to find the LCM of three or more natural numbers. Another, more efficient, methodfor finding the LCM of several numbers is the prime factorization method. For example, tofind first find the prime factorizations of 40 and 12, namely, and respectively.

22 # 3,23 # 5LCM140, 122,

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If then m is a multiple of 40 and must contain both and 5 as fac-tors. Also, m is a multiple of 12 and must contain and 3 as factors. Since is a multipleof then In general, we have the following:

To find the LCM of two natural numbers, first find the prime factorization of each number. Thentake each of the primes that are factors of either of the given numbers. The LCM is the productof these primes, each raised to the greatest power of the prime that occurs in either of the primefactorizations.

m = 23 # 5 # 3 = 120.22,232223m = LCM140, 122,

Figure 5-31

Find the LCM of 2520 and 10,530.

Solution

LCM12520, 105302 = 23 # 34 # 5 # 7 # 13 = 294,840 10,530 = 2 # 34 # 5 # 13.

2520 = 23 # 32 # 5 # 7.

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-30

▲

▲

The prime factorization method can also be used to find the LCM of more than twonumbers. For example, to find LCM(12, 108, 120), we can proceed as follows:

Then,

The GCD-LCM Product Method

To see the connection between the GCD and LCM, consider the GCD and LCM of 24and 30. The prime factorizations of these numbers are

A diagram showing the prime factorization is given in Figure 5-31.

30 = 2 # 3 # 5 24 = 23 # 3

LCM112, 108, 1202 = 23 # 33 # 5 = 1080.

120 = 23 # 3 # 5 108 = 22 # 33

12 = 22 # 3

22

23

5

Prime Factors of 24 Prime Factors of 30

Notice that is the product of the factors in the shaded region, andis the product of the factors in the combined regions. Also no-

tice that

This shows that the product of the GCD and LCM of 24 and 30 is equal to In gen-eral, the connection between their GCD and LCM of any pair of natural numbers is givenby Theorem 5–29.

24 # 30.

GCD124, 302 # LCM124, 302 = 12 # 32123 # 3 # 52 = 123 # 3212 # 3 # 52 = 24 # 30

LCM124, 302 = 23 # 3 # 5GCD124, 302 = 2 # 3

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Theorem 5–29 can be justified in several ways. Here is a specific example that suggestshow the theorem might be proved. Suppose

Then,

Now we have

GCD1a, b2 = 510 # 720 # 114LCM1a, b2 = 513 # 725 # 116 # 13 and

b = 510 # 725 # 116 # 13a = 513 # 720 # 114 and

Theorem 5–29For any two natural numbers a and b,

GCD1a, b2 # LCM1a, b2 = ab

Find .

Solution By the Euclidean algorithm, By Theorem 5–29,

Consequently,

LCM1731, 9522 =731 # 952

17= 40,936

17 # LCM1731, 9522 = 731 # 952

GCD1731, 9522 = 17.

LCM1731, 9522

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-31

▲

▲

The Division-by-Primes Method

Another procedure for finding the LCM of several natural numbers involves division byprimes. For example, to find , we start with the least prime that divides atleast one of the given numbers and divide as follows:

Because 2 does not divide 75, simply bring down the 75. To obtain the LCM using this pro-cedure, continue the division process until the row of answers consists of relatively primenumbers as shown next.

2:12, 75, 1206, 75, 60

LCM112, 75, 1202

ab = 513+10 # 720+25 # 114+6 # 13 LCM1a, b2 # GCD1a, b2 = 513+10 # 725+20 # 116+4 # 13 and

For the preceding values of a and b, Theorem 5–29 is true. Notice, however, that in theproduct we have all the powers of the primes appearing in a or in b,because for the LCM we take the greater of the powers of the common primes and for theGCD the lesser. Also in ab we have all the powers. Hence, Theorem 5–29 is true in general.

The Euclidean Algorithm Method

Theorem 5–29 is useful for finding the LCM of two numbers a and b when their prime fac-torizations are not easy to find. GCD(a, b) can be found by the Euclidean algorithm, theproduct ab can be found by simple multiplication, and can be found by division.LCM1a, b2

LCM1a, b2 # GCD1a, b2

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Thus, LCM112, 75, 1202 = 2 # 2 # 2 # 3 # 5 # 1 # 5 # 1 = 23 # 3 # 52 = 600.

2 12, 75, 1202 6, 75, 602 3, 75, 303 3, 75, 155 1, 25, 5 1, 5, 1

Assessment 5-5A

1. Find the GCD and the LCM for each of the followingusing the intersection-of-sets method:a. 18 and 10 b. 24 and 36c. 8, 24, and 52 d. 7 and 9

2. Find the GCD and the LCM for each of the followingusing the prime factorization method:a. 132 and 504 b. 65 and 1690c. 900, 96, and 630 d. 108 and 360

3. Find the GCD for each of the following using the Eu-clidean algorithm:a. 220 and 2924 b. 14,595 and 10,856

4. Find the LCM for each of the following using anymethod:a. 24 and 36b. 72 and 90 and 96c. 90 and 105 and 315d. and

5. Find the LCM for each of the following pairs of numbersusing Theorem 5–29 and the answers from problem 3:a. 220 and 2924b. 14,595 and 10,856

6. Use colored rods to find the GCD and the LCM of 6and 10.

7. In Quinn’s dormitory room, there are three snooze-alarm clocks, each of which is set at a different time.Clock A goes off every 15 min, clock B goes off every40 min, and clock C goes off every 60 min. If all threeclocks go off at 6:00 A.M., answer the following:a. How long will it be before the clocks go off simulta-

neously again after 6:00 A.M.?b. Would the answer to (a) be different if clock B went off

every 15 min and clock A went off every 40 min?8. Midas has 120 gold coins and 144 silver coins. He wants

to place his gold coins and his silver coins in stacks sothat there are the same number of coins in each stack.What is the greatest number of coins that he can placein each stack?

251009100

9. By selling cookies at 24¢ each, José made enough moneyto buy several cans of pop costing 45¢ per can. If he hadno money left over after buying the pop, what is the leastnumber of cookies he could have sold?

10. Two bike riders ride around in a circular path. The firstrider completes one round in 12 min and the secondrider completes it in 18 min. If they both start at thesame place and the same time and go in the same direc-tion, after how many minutes will they meet again at thestarting place?

11. Three motorcyclists ride around a circular race coursestarting at the same place and the same time. The firstpasses the starting point every 12 min, the second every18 min, and the third every 16 min. After how manyminutes will all three pass the starting point again at thesame time? Explain your reasoning.

12. Assume a and b are natural numbers and answer thefollowing:a. If find b. Find and c. Find and d. If find and

13. Classify each of the following as true or false:a. If then a and b cannot both be even.b. If then both a and b are even.c. If a and b are even, then

14. To find , it is possible to find, which is 4, and then find

which is 4. Use this approach and the Euclidean algo-rithm to finda. .b. .

15. Show that 97,219,988,751 and 4 are relatively prime.16. The radio station gave away a discount coupon for every

twelfth and thirteenth caller. Every twentieth caller re-ceived free concert tickets. Which caller was first to getboth a coupon and a concert ticket?

GCD134578, 4618, 46192GCD1120, 75, 1052

GCD14, 122,GCD124, 202GCD124, 20, 122

GCD1a, b2 = 2.GCD1a, b2 = 2,GCD1a, b2 = 1,

LCM1a, b2.GCD1a, b2a ƒ b,LCM1a2, a2.GCD1a2, a2

LCM1a, a2.GCD1a, a2LCM1a, b2.GCD1a, b2 = 1,

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17. Jackie spent the same amount of money on DVDs thatshe did on compact discs. If DVDs cost $12 and CDs$16, what is the least amount she could have spent oneach?

18. At the Party Store, paper plates come in packages of 30,paper cups in packages of 15, and napkins in packages of20. What is the least number of plates, cups, and napkinsthat can be purchased so that there is an equal numberof each?

19. Diagrams can be used to show factors of two or morenumbers. Draw diagrams to show the prime factors foreach of the following sets of three numbers:a. 10, 15, 60 b. 8, 16, 24

20. What are the factors of 21. In algebra it is often necessary to factor an expression

as much as possible. For example, a3b2 + a2b3 =

410?

and no further factoring is possible withoutknowing the values of a and b. Notice that is theGCD of and . Factor each of the following asmuch as possible:a.b.

22. Label the following statements as “always true,” “some-times true,” or “never true.” Justify your answers.a.b.

23. Find the GCD and the LCM of each of the following.(Do not compute the products.)a. 10!, 11!b. 10!,

24. Factor 1 billion into a product of two numbers, neither ofwhich contains any zeros.

10! + 1

GCD1-a, b2 = GCD1a, -b2 = GCD1-a, -b2GCD1a, b2 = GCD1 ƒ a ƒ , b2 = GCD1 ƒ a ƒ , ƒ b ƒ 2

12x3y2z2 + 18x2y4z3 + 24x4y3z412x4y3 + 18x3y4

a2b3a3b2a2b2

a2b21a + b2

Assessment 5-5B

1. Find the GCD and the LCM for each of the followingusing the intersection-of-sets method:a. 12 and 18 b. 18 and 36c. 12, 18, and 24 d. 6 and 11

2. Find the GCD and the LCM for each of the followingusing the prime factorization method:a. 11 and 19 b. 140 and 320c. 800, 75, and 450 d. 104 and 320

3. Find the GCD for each of the following using the Eu-clidean algorithm:a. 14,560 and 8250 b. 8424 and 2520

4. Find the LCM for each of the following using anymethod:a. 25 and 36b. 82 and 90 and 50c. 80 and 105 and 315d. and

5. Find the LCM for each of the following pairs of numbersusing Theorem 5–29 and the answers from problem 3:a. 14,560 and 8250b. 8424 and 2520

6. A movie rental store gives a free popcorn to every fourthcustomer and a free movie rental to every sixth customer.Use the number-line method to find which customer wasthe first to win both prizes.

7. Use colored rods to find the GCD and the LCM of 4and 10.

8. Bill and Sue both work at night. Bill has every sixthnight off and Sue has every eighth night off. If they areboth off tonight, how many nights will it be before theyare both off again?

9. Bijous I and II start their movies at 7:00 P.M. The movie atBijou I takes 75 min, while the movie at Bijou II takes

501008100

90 min. If the shows run continuously, when will theystart at the same time again?

10. A rectangular field with dimensions 75 ft by 625 ft is tobe divided into same-size square plots. If the sides of thesquares need to be whole numbers of feet long, what area. the largest squares possible and how many such squares

will fit in the field?b. what are the smallest squares possible?c. what other size squares are possible?

11. The principal of Valley Elementary School wants to di-vide each of the three fourth-grade classes into smallsame-size groups with at least 2 students in each. If theclasses have 18, 24, and 36 students, respectively, whatsize groups are possible?

12. Assume a and b are natural numbers and answer thefollowing:a. If a and b are two primes, find and

.b. What is the relationship between a and b if

c. What is the relationship between a and b if

13. Classify each of the following as true or false: for all nat-ural numbers a and b. a.b. For all natural numbers a and b, c.d.

14. To find GCD(24, 20, 12), it is possible to find GCD(24,20), which is 4, and then find GCD(4, 12), which is 4.Use this approach and the Euclidean algorithm to finda. .b. GCD15284, 1250, 12802.

GCD1180, 240, 3062

LCM1a, b2 Ú a.GCD1a, b2 … a.

LCM1a, b2 ƒ ab.1a, b2.LCM1a, b2 ƒ GCD

LCM1a, b2 = a?

GCD1a, b2 = a?

LCM1a, b2GCD1a, b2

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15. Show that 181,345,913 and 11 are relatively prime.16. Larry and Mary bought a special 360-day joint member-

ship to a tennis club. Larry will use the club every otherday, and Mary will use the club every third day. Theyboth use the club on the first day. How many days willneither person use the club in the 360 days?

17. Determine how many complete revolutions gear 2 in thefollowing must make before the arrows are lined upagain.

19. Diagrams can be used to show factors of two or morenumbers. Draw diagrams to show the prime factors foreach of the following sets of three numbers:a. 12, 14, 70 b. 6, 8, 18

20. Find all natural numbers x such that and

21. In algebra it is often necessary to factor an expres-sion as much as possible. For example,

and no further factoring is possible withoutknowing the values of a and b. Notice that is theGCD of and Factor each of the following asmuch as possible:a.b.

22. Find all values of a and b for which the following aretrue.a. If then b. If then

23. Find the GCD and the LCM of each of the following.(Do not compute the products.)a. pqr, qrs (where p, q, r, s are prime numbers)b.

24. If you find the sum of any two-digit number and the num-ber formed by reversing its digits, the resulting number isalways divisible by which three positive integers?

210, 28

GCD1a, b, c2 = 1.LCM1a, b, c2 = abc,GCD1a2, b22 = GCD1a, b32.GCD1a, b2 = 1,

61x2 - y22 + 121x2 - y22 + 181 y2 - x22

61x2 - y22 - 31x - y2 + 91 y - x2

a2b3.a3b2a2b2

a2b21a + b2a3b2 + a2b3 =

1 … x … 25.GCD125, x2 = 1

Gear 1 Gear 2

48 teeth

28 teeth

18. Determine how many complete revolutions each gear inthe following must make before the arrows are lined upagain:

Gear 1Gear 2

Gear 3

40 teeth24 teeth

60 teeth


Communication1. Can two natural numbers have a greatest common mul-

tiple? Explain your answer.2. Describe to a sixth-grade student the difference between

a divisor and a multiple.3. Is it true that Ex-

plain your answer.4. A rectangular plot of land is 558 m by 1212 m. A surveyor

needs to divide the plot into the largest possible squareplots of the same size, being a whole number of meterslong. What is the size of each square and how manysquare plots can be created? Explain your reasoning.

5. Suppose that Is it necessarily true thatExplain your reasoning.

6. Suppose Does that alwaysimply that Justify your answer.GCD1a, b, c2 = 2?

GCD1a, b2 = GCD1b, c2 = 2.GCD1a, b2 = GCD1b, c2 = 1?

GCD1a, b, c2 = 1.

GCD1a, b, c2 # LCM1a, b, c2 = abc?

7. How can you tell from the prime factorization of twonumbers if their LCM equals the product of the num-bers? Explain your reasoning.

8. Can the LCM of two positive numbers ever be greaterthan the product of the numbers? Explain your rea-soning.

9. Let and Jackie con-jectures that for all integers m andn. Check Jackie’s conjecture for three different pairs ofintegers.

Open-Ended10. Make up a word problem that can be solved by finding

the GCD and another that can be solved by finding theLCM. Solve your problems and explain why you aresure that your approach is correct.

GCD1m + n, l2 = gLCM1m, n2 = l.GCD1m, n2 = g

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11. Find three pairs of numbers for which the LCM of thenumbers in a pair is smaller than the product of the twonumbers.

12. Describe infinitely many pairs of numbers whose GCD isa. 2b. 6c. 91

Cooperative Learning13. Each member of your group should examine a different

elementary-school book that covers GCDs and LCMs.Report to the group on what methods were used and howthey were used.

14. a. In your group, discuss whether the Euclidean algo-rithm for finding the GCD of two numbers should beintroduced in middle school (to all students? to some?).Why or why not?

b. If you decide that it should be introduced in middleschool, discuss how it should be introduced. Reportyour group’s decision to the class.

Questions from the Classroom15. Alba asked why we don’t talk about the LCD (least com-

mon divisor) and GCM (greatest common multiple).How do you respond?

16. A student says that for any two natural numbers a and b,GCD(a, b) divides LCM(a, b) and, hence,

. Is the student correct? Why or why not?17. A student asks about the relation between least common

multiple and least common denominator. How do yourespond?

18. A student wants to know how many integers between 1and 10,000 inclusive are either multiples of 3 or multiplesof 5. She wonders if it is correct to find the number ofthose integers that are multiples of 3 and add the numberof those that are multiples of 5. How do you respond?

LCM1a, b2GCD1a, b2 6

19. Dolores claims that she has a shortcut for finding theGCD using the Euclidean algorithm. She says when theremainder is large she uses a negative “remainder.” Forexample, to find GCD(2132, 534), she divides 2132 by534 and gets which gives remain-der 530. In such case, she writes andclaims that

Is the approach correct and if so, why?

Review Problems20. Find two whole numbers x and y such that

and neither x nor y contains any zeros as digits.21. Fill each blank space with a single digit that makes the

corresponding statement true. Find all possible answers.a.b.c.

22. Is 3111 a prime? Prove your answer.23. Find a number that has exactly six prime factors.24. Produce the least positive number that is divisible by 2,

3, 4, 5, 6, 7, 8, 9, 10, and 11.25. What is the greatest prime that must be used to deter-

mine if 2089 is prime?


The least common multiple of 8, 12, and a third number is120. Which of the following could be the third number?a. 15 b. 16 c. 24 d. 32 e. 48NAEP, Grade 8, 1990

23 ƒ 103_611 ƒ 8_6913 ƒ 83_51

x y = 1,000,000

= 2 1because 4�5342 = GCD14, 5342

GCD12132, 5342 = GCD1-4, 5342

2132 = 4 # 534 - 42132 = 3 # 534 + 530,

TECHNOLOGY CORNER

1. Write a spreadsheet to generate the first 50 multiples of 3 and the first 50 multiples of 4. Describe theintersection of the two sets.

2. Use a spreadsheet to find the factors of 2486. How far down do you need to copy the formula to besure you have found all the divisors?

3. Make a spreadsheet with four columns:Column A—the multiples of 6Column B—the multiples of 9Column C—the multiples of 12Column D—the multiples of 15a. What is the least number that appears in all four columns?b. Explain how to find this number without using a spreadsheet.

A B

1 1 = 2486>A12 23 3

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12 12

3

456

11

7

10

8

9

Figure 5-33

*5-6

Section 5-6 Clock and Modular Arithmetic 329

Clock and Modular Arithmetic

In this section, we investigate clock arithmetic. Consider the following:

a. A doctor’s prescription says to take a pill every 8 hr. If you take the first pill at 7:00 A.M.,when should you take the next two pills?

b. Suppose you are following a bean soup recipe that calls for letting the beans soak for12 hr. If you begin soaking them at 8:00 P.M., when should you take them out?

c. The odometer on a car gives the total miles traveled up to 99,999 mi and then starts count-ing from 0. If the odometer shows 99,124 mi, what will it show after a trip of 2,116 mi?

Some of these situations involve the ability to solve arithmetic problems using clocks. Mostpeople can solve these problems without thinking much about what they are doing. It ispossible to use the clock in Figure 5-33 to determine that 8 hr after 7:00 A.M. is 3:00 P.M.and 8 hr after that is 11:00 P.M. Also, 12 hr after 8:00 P.M. is 8:00 A.M. We could record theseadditions on the clock as

where indicates addition on a 12-hr clock.�

7 � 8 = 3, 3 � 8 = 11, 8 � 12 = 8

You probably noticed the special role of 12 when you found that In the 12-hrclock arithmetic, 12 acts like a 0 if you were adding in the set of whole numbers. An additiontable for the finite system based on the clock is shown in Table 5-4.

8 � 12 = 8.

BRAIN TEASER For any rectangle such that find a rule for determining thenumber of unit squares that a diagonal passes through. For example, in the drawings in Figure 5-32,the diagonal passes through 8 and 6 unit squares, respectively.

11 * 12GCD1n, m2 = 1,n * m

(a) (b)

Figure 5-32

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When we allow numbers other than those on the 12-hr clock to be added, such aswe find that numbers such as act like 12 (or 0). Likewise, the

numbers act like the number 1. Similarly, we can generate classes of numbersthat act like each of the numbers on the 12-hr clock. The members of any one class differby multiples of 12. Consequently, to perform additions on a 12-hr clock, we perform regu-lar addition, divide by 12, and record the remainder as the answer. For example, we can find

and as follows:

Next divide The quotient is 1 with a remainder of 7. Therefore,

Next divide The quotient is 1 with a remainder of 8. Therefore,

Clock multiplication can be defined using repeated addition, as with whole numbers.For example, where denotes clock multiplication. Similarly,

We could also find as has re-mainder 4, so Likewise, we could find as has remainder 3,so This leads to the following definition.3 � 5 = 3.

13 # 52 , 123 � 52 � 8 = 4.12 # 82 , 122 � 83 � 5 = 15 � 52� 5 = 10 � 5 = 3.

�2 � 8 = 8 � 8 = 4,

8 � 12 = 8.20 , 12.8 + 12 = 20.

11 � 8 = 7.19 , 12.11 + 8 = 19.

8 � 1211 � 8

13, 25, 37, Á24, 36, 48, Á8 � 24 = 8,

Definition12-Hr Clock Sums and Products To compute a sum or product in 12-hr clock arithmetic,perform the computation as with whole numbers, divide by 12, and take the remainder as theanswer.

NOW TRY THIS 5-18 Examine Table 5-4 to determine if the following properties hold for on theset of numbers in the table:

a. Commutative property of additionb. Identity property of additionc. Inverse property of addition

�

� 12 1 2 3 4 5 6 7 8 9 10 11

12 12 1 2 3 4 5 6 7 8 9 10 111 1 2 3 4 5 6 7 8 9 10 11 122 2 3 4 5 6 7 8 9 10 11 12 13 3 4 5 6 7 8 9 10 11 12 1 24 4 5 6 7 8 9 10 11 12 1 2 35 5 6 7 8 9 10 11 12 1 2 3 46 6 7 8 9 10 11 12 1 2 3 4 57 7 8 9 10 11 12 1 2 3 4 5 68 8 9 10 11 12 1 2 3 4 5 6 79 9 10 11 12 1 2 3 4 5 6 7 8

10 10 11 12 1 2 3 4 5 6 7 8 911 11 12 1 2 3 4 5 6 7 8 9 10

Table 5-4

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Section 5-6 Clock and Modular Arithmetic 331▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-32

To perform other operations on the clock, such as where denotes clock sub-traction, we could interpret it as the time 9 hr before 2 o’clock. Counting backward (counter-clockwise) 9 units from 2 reveals that If subtraction on the clock is defined in termsof addition, we have if, and only if, Consequently,

Clock division can be defined in terms of multiplication. For example, 8 s4 where s4

denotes clock division, if, and only if, for a unique x in the set Because and 8 is unique, we have then 8 s4 5 = 4.5 � 4 = 8

51, 2, 3, Á , 126.8 = 5 � x5 = x,

x = 5.2 = 9 � x.2 � 9 = x2 � 9 = 5.

�2 � 9,

Perform each of the following computations on a 12-hr clock:

a. b.c. d.

Solution a. has remainder 4. Hence, b. since by counting forward or backward 12-hr, you arrive at the

original position.c. This should be clear from looking at the clock, but it can also be

found by using the definition of subtraction in terms of addition.d. because 8 � 8 = 4.4 � 8 = 8

4 � 4 = 12.

4 � 12 = 4,8 � 8 = 4.18 + 82 , 12

4 � 84 � 44 � 128 � 8

▲

▲

Adding or subtracting 12 on a 12-hr clock leaves a number unchanged. Thus 12 behavesas 0 does in integer addition or subtraction and is the additive identity for addition on the12-hr clock. Similarly, on a 5-hr clock 5 behaves as 0 does.

Addition, subtraction, and multiplication on a 12-hr clock can be performed for any twonumbers but, as shown in Example 5-33(d), not all divisions can be performed. Division by 12,the additive identity, on a 12-hr clock either can never be performed or is not meaningful, sinceit does not yield a unique answer. However, there are clocks on which all divisions can beperformed, except by the corresponding additive identities. One such clock is a 5-hr clock,shown in Figure 5-34.

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-33

▲

▲

Perform the following operations on a 12-hr clock, if possible:

a. b. 2 s4 7c. 3 s4 2 d. 5 s4 12

Solution a. or has remainder 9.Hence,

b. 2 s4 if, and only if, and x is unique. Consequently, c. 3s4 if, and only if, and x is unique. Multiplying each of the

numbers 12 by 2 shows that none of the multiplications yields3. Thus, the equation has no solution, and consequently, 3 s4 2is undefined.

d. 5 s4 if, and only if, and x is unique. However, for every x in the set Thus, has no solution onthe clock, and therefore 5 s4 12 is undefined.

5 = 12 � x51, 2, 3, 4, Á , 126.12 � x = 125 = 12 � x12 = x

3 = 2 � x1, 2, 3, 4, Á ,

3 = 2 � x2 = xx = 2.2 = 7 � x7 = x

3 � 11 = 9.13 # 112 , 123 � 11 = 111 � 112� 11 = 10 � 11 = 9

3 � 11

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S M T W T F S

1

8

15

22

29

2

9

16

23

30

3

10

17

24

4

11

18

25

5

12

19

26

APRIL

6

13

20

27

7

14

21

28

Figure 5-35

Definition of Modular CongruenceFor integers a and b, a is congruent to b modulo m, written (mod m), if, and only if,

is a multiple of m, where m is a positive integer greater than 1.a - ba K b

R E M A R K This definition could be written as (mod m) if, and only if, where m is a positive integer greater than 1.

m ƒ 1a - b2,a K b

5

4

3 2

1

Figure 5-34

On this clock, 3 s4 Since adding 5to any number yields the original number, 5 is the additive identity for this 5 hr clock, asseen in Table 5-5(a). Consequently, you might suspect that division by 5 is not possibleon a 5-hr clock. To determine which divisions are possible, consider Table 5-5(b), amultiplication table for 5-hr clock arithmetic. To find 1 s4 2, we write 1 s4 2 = x,which is equivalent to The second row of Table 5-5(b) shows that

The unique solutionof is so 1 s4 2 The information given in the second row of thetable can be used to determine the following divisions:

2 s4

3 s4

4 s4

5 s4

Because every element occurs in the second row, division by 2 is always possible. Similarly,division by all other numbers, except 5, is always possible. In the problem set, you are askedto perform arithmetic on different clocks and to investigate for which clocks all computa-tions, except division by the additive identity, can be performed.

Modular ArithmeticMany of the concepts for clock arithmetic can be used to work problems that involve a cal-endar. On the calendar in Figure 5-35, the five Sundays have dates 1, 8, 15, 22, and 29. Anytwo of these dates for Sunday differ by a multiple of 7. The same property is true for anyother day of the week. For example, the second and thirtieth days fall on the same day, since

and 28 is a multiple of 7. We say that 30 is congruent to 2, modulo 7, and wewrite (mod 7). Similarly, because 18 and 6 differ by a multiple of 12, we write

(mod 12). This is generalized in the following definition.18 K 630 K 2

30 - 2 = 28

2 = 5 because 5 = 2 � 52 = 2 because 4 = 2 � 22 = 4 because 3 = 2 � 42 = 1 because 2 = 2 � 1

= 3.x = 3,1 = 2 � x1, 2 � 4 = 3, and 2 � 5 = 5. 2 � 2 = 4, 2 � 3 =2 � 1 = 2,

1 = 2 � x.

4 = 2.3 � 4 = 2, 2 � 3 = 4, 2 � 4 = 3, and

Table 5-5

� 1 2 3 4 5

1 2 3 4 5 12 3 4 5 1 23 4 5 1 2 34 5 1 2 3 45 1 2 3 4 5

(a)

� 1 2 3 4 5

1 1 2 3 4 52 2 4 1 3 53 3 1 4 2 54 4 3 2 1 55 5 5 5 5 5

(b)

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Notice that 18 and 25 are congruent modulo 7 and each number leaves the same remain-der, 4, upon division by 7. Indeed, and In general, we havethe following property: Two whole numbers are congruent modulo m if, and only if, their remain-ders on division by m are the same.

25 = 3 # 7 + 4.18 = 2 # 7 + 4

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼Example 5-34

▲

▲

Tell why each of the following is true:

a. (mod 10) b. (mod 4) c. (mod 7)d. (mod 11) e. (mod m)

Solution a. (mod 10) because is a multiple of 10, or because 23 and 3 leavethe same remainder, 3, upon division by 10.

b. (mod 4) because is a multiple of 4.c. (mod 7) because is not a multiple of 7.d. (mod 11) because is a multiple of 11.e. (mod m) because is a multiple of m, or because m and 0 have the

same remainder, 0, upon division by m.m - 0m K 0

10 - 1-12 = 1110 K -123 - 323 [ 323 - 323 K 3

23 - 323 K 3

m K 010 K -123 [ 323 K 323 K 3

R E M A R K Example 5-34(e) shows that m behaves like 0 modulo m. This is also evidentfor in Table 5-4 and for in Table 5-5.m = 5m = 12

Find all integers x such that (mod 10).

Solution (mod 10) if, and only if, where k is any integer. Consequently,Letting yields the sequence . Likewise,

letting yields the negative integers Thetwo sequences can be combined to give the solution set

5Á , -39, -29, -19, -9, 1, 11, 21, 31, 41, 51, Á 6

-9, -19, -29, -39, Á .k = -1, -2, -3, -4, Á1, 11, 21, 31, 41, Ák = 0, 1, 2, 3, Áx = 10k + 1.

x - 1 = 10k,x K 1

x K 1

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-35

▲

▲

The button on a calculator can be used to work with modular arithmetic. If wepress the following sequence of buttons, we see that (mod 9) because the remain-der when 4325 is divided by 9 is 5:

and the display shows a remainder of 5.

=9INT ,5234

4325 K 5INT ,

Heidi signed a promissory note that will become due in 90 days. She is worried that it willbecome due on a weekend. She signed the note on a Monday. On what day of the week willit be due?

Solution Because we know that (mod 7). On a fraction calculator,you could enter , and a quotient of 12 with remainder 6 would be dis-played. Therefore, the note will come due 12 wk and 6 days after Monday, which is a Sunday.

=7INT ,0990 K 690 = 7 # 12 + 6,

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-36

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Assessment 5-6B

1. The Smiths left on a car trip at 6:00 A.M. They traveledfor exactly 15 hr. When did they arrive?

2. Perform each of the following operations on a 12-hrclock, if possible:a. b. c. 4 � 65 � 116 � 6

d. e. f.g. 2 s4 3 h. 4 s4 6

3. Perform each of the following operations on a 5-hr clock:a. b. c.d. e. f.g. 2 s4 4 h. 4 s4 4

5 � 33 � 31 � 34 � 42 � 24 � 5

3 � 34 � 95 � 8

a. If it is now Monday, October 14, on what day of the week will October 14 fall next yearif next year is not a leap year?

b. If Christmas falls on Thursday this year, on what day of the week will Christmas fall nextyear if next year is a leap year?

Solution a. Because next year is not a leap year, we have 365 days in the year. Becausewe have (mod 7). Thus, 365 days after October 14

will be 52 wk and 1 day later. Thus, October 14 will be on a Tuesday.b. There are 366 days in a leap year, and (mod 7). Thus, Christmas will

be 2 days after Thursday, on Saturday.366 K 2

365 K 1365 = 52 # 7 + 1,

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

Example 5-37

▲

▲

Assessment 5-6A

1. Dr. Harper prescribed some medicine for Camile. Sheis supposed to take a dose every 6 hr. If she takes herfirst dose at 8:00 A.M., when should she take her nextdose?

2. Perform each of the following operations on a 12-hrclock, if possible:a. b. c.d. e. f.g. 1 s4 3 h. 2 s4 5

2 � 23 � 94 � 83 � 94 � 107 � 8

c. d.e. f.

7. a. If April 23 falls on Tuesday, what are the dates of theother Tuesdays in April of that year?

b. If July 2 falls on Tuesday, list the dates of the Wednes-days in July.

c. If September 3 falls on Monday, on what day of theweek will it fall next year if next year is a leap year?

8. Fill in each of the following blanks so that the answer isnonnegative and the least possible number:a. (mod 5)b. (mod 3)c. (mod 11)d. (mod 10)

9. a. Find all x such that (mod 2).b. Find all x such that (mod 2).c. Find all x such that (mod 5).

10. A new clock is started on Sunday at 10:00 P.M. If theclock continues running nonstop, on what day and hourrounded to the nearest hour will it be when the clockreaches the 100,000th second?

11. If the following pattern continues,

what will be the 101st letter in the pattern?

CLOCK CLOCK CLOCK CLOCK Á

x K 3x K 1x K 0

-23 K _____3498 K _____3498 K _____29 K _____

1-22� 1-221-22� 1-32-12 � 221-22� 1-22

3. Perform each of the following operations on a 5-hr clock:a. b. c.d. e. f.g. 3 s4 4 h. 1 s4 4

4. a. Construct an addition table for a 9-hr clock.b. Using the addition table in (a), find and c. Using the definition of subtraction in terms of addi-

tion, show that subtraction can always be performedon a 9-hr clock.

5. a. Construct a multiplication table for a 9-hr clock.b. Use the multiplication table in (a) to find 3 s4 5 and

4 s4 6.c. Use the multiplication table to find whether division

by numbers different from 9 is always possible.6. On a 5-hr clock, find each of the following:

a. Additive inverse of 2 b. Additive inverse of 3

2 � 5.5 � 6

2 � 34 � 41 � 43 � 43 � 33 � 4

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4. a. Construct an addition table for a 7-hr clock.b. Using the addition table in (a), find and c. Using the definition of subtraction in terms of addi-

tion, show that subtraction can always be performedon a 7-hr clock.

5. a. Construct a multiplication table for a 7-hr clock.b. Use the multiplication table in (a) to find 3 s4 5 and

4 s4 6.c. Use the multiplication table to find whether division

by numbers different from 7 is always possible.6. On a 12-hr clock, find each of the following:

a. Additive inverse of 2 b. Additive inverse of 3c. d.e. f.

7. a. If April 8 falls on Friday, what are the dates of theother Fridays in April?

b. If July 4 falls on Tuesday, what day will it fall on nextyear if next year is not a leap year?

1-22� 1-321-22� 1-32-12 � 321-22� 1-32

2 � 5.5 � 6c. Do the 125th day and the 256th day of the year fall

on the same day of the week? Explain why.8. Fill in each of the following blanks so that the answer is

nonnegative and the least possible number:a. (mod 3)b. (mod 5)c. (mod 10)d. (mod 11)

9. a. Find all x such that (mod 3).b. Find all x such that (mod 3).c. Find all x such that (mod 7).

10. Continue a possible pattern by listing the next four termsin each sequence in clock arithmetic.a.b. 3, 8, 13, 4, 9, 14, 5, 10, 1, Á

3, 8, 1, 6, 11, 4, 9, 2, 7, Á

x K 3x K 1x K 0

-23 K _____3498 K _____3498 K _____29 K _____


Communication1. Explain or find each of the following:

a. A number congruent modulo 10 to the numberformed by its last digit

b. The last digit of c. A number congruent modulo 100 to the number

formed by its last two digits2. a. For all a, b, c, and d, explain why

(mod 9).b. If (mod m), what is m? Ex-

plain your reasoning.

Open-Ended3. On a clock we define the additive inverse of a in the same

way as the additive inverse was defined for integers. Hav-ing this definition in mind, list some similarities andsome differences between the number system on theclock and the set of integers. Justify your answers.

Cooperative Learning4. a. Have members of your group construct the multipli-

cation tables for 3 hr, 4 hr, 6 hr, and 11 hr clocks.

abcdfive K a + b + c + d

abcd K a + b + c + d

2180 - 1

b. Compare your results. On which of the clocks in (a)can divisions by numbers other than the additiveidentity always be performed?

c. How do the multiplication tables of clocks for whichdivision can always be performed (except by an addi-tive identity) differ from the multiplication tables ofclocks for which division is not always meaningful?

Questions from the Classroom5. Dan was trying to see how fractions like might work in

a 5 hr clock system. He said that he showed that 1 s4 4 isgreater than 3. He wants to know if this could possiblybe correct. How do you respond?

6. Ally wants to know what elements of a 5 hr clock systemhave a multiplicative inverse. What do you tell her?

7. Zita claims that on the 5 hr clock shown in Figure 5-34there is no 0 so there can be no additive identity. How doyou respond?

14

BRAIN TEASER How many primes are in the following sequence?

9, 98, 987, 9876, Á , 987654321, 9876543219, 98765432198, Á

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Chapter Outline

I. Basic concepts of integersA. The set of integers, I, is

B. The distance from any integer to 0 is theabsolute value of the integer. The absolutevalue of an integer x is denoted If then and if then

C. Operations with integers1. Addition: For any integers a and b,

2. Subtractiona. For all integers a and b, then

if, and only if, .b. For all integers a and

3. Multiplication: For any integers a and b,a.b.

4. Division: If a and b are any integers withthen is the unique integer c, if

it exists, such that .5. Order of operations: When addition, sub-

traction, multiplication, and division appearwithout parentheses, multiplications and di-visions are done first in the order of theirappearance from left to right and then addi-tions and subtractions are done in the orderof their appearance from left to right. Anyarithmetic in parentheses is done first.

II. The system of integersA. The set of integers, along with the operations

of addition and multiplication, satisfy the fol-lowing properties:

a = bca , bb Z 0,

1-a2b = b1-a2 = -1ab2.1-a21-b2 = ab.

a + -b.b, a - b =

a = b + na - b = n

-a + -b = -1a + b2

ƒ x ƒ = -x.x 6 0,ƒ x ƒ = xx Ú 0,ƒ x ƒ .

0, 1, 2, 3, Á 6.5Á , -3, -2, -1,

B. Zero multiplication property of integers Forany integer

C. For all integers a, b, and c,1.2.3. (difference-of-

squares formula).III. Divisibility

A. If a and b are any integers, then b divides a, de-noted , if, and only if, there is a unique inte-ger c such that .

B. The following are basic divisibility theoremsfor integers a, b, and d:1. If and k is any integer, then .2. If and , then and 3. If and then and

C. Divisibility tests1. An integer is divisible by 2, 5, or 10 if, and

only if, its units digit is divisible by 2, 5, or10, respectively.

2. An integer is divisible by 4 if, and only if,the last two digits of the integer represent anumber divisible by 4.

3. An integer is divisible by 8 if, and only if,the last three digits of the integer representa number divisible by 8.

4. An integer is divisible by 3 or 9 if, and onlyif, the sum of its digits is divisible by 3 or 9,respectively.

5. An integer is divisible by 11 if, and only if,the sum of the digits in the places that areeven powers of 10 minus the sum of thedigits in the places that are odd powers of10 is divisible by 11.

6. An integer is divisible by 6 if, and only if,the integer is divisible by both 2 and 3.

IV. Prime and composite numbersA. Positive integers that have exactly two positive

divisors are primes. Integers greater than 1that are not primes are composites.

B. Fundamental Theorem of Arithmetic: Everycomposite number has one, and only one, primefactorization, aside from variation in the orderof the prime factors.

d�1a - b2.d�1a + b2d�b,d ƒ a

d ƒ 1a - b2.d ƒ 1a + b2d ƒ bd ƒ ad ƒ kad ƒ a

a = cbb ƒ a

1a + b21a - b2 = a2 - b2a - 1b - c2 = a - b + c.-1-a2 = a.

a, a # 0 = 0 = 0 # a.

Property + *

Closure Yes YesCommutative Yes YesAssociative Yes YesIdentity Yes, 0 Yes, 1Inverse Yes No

Distributive Property of Multiplication over Addition

Hint for Solving the Preliminary ProblemThe difference-of-squares formula along with the work done on the Gauss problem

in Chapter 1 will help solve this problem.

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C. Criterion for determining if a given number nis prime: If n is not divisible by any prime p suchthat then n is prime.

D. If the prime factorization of a number is ,where p and q are prime, then the number ofdivisors of n is

V. Greatest common divisor and least common multipleA. The greatest common divisor (GCD) of two

or more natural numbers is the greatest divisor,or factor, that the numbers have in common.

B. Euclidean algorithm: If a and b are positive inte-gers and then where r is the remainder when a is divided by b.The procedure of finding the GCD of two num-bers a and b by using this result repeatedly is theEuclidean algorithm.

GCD1a, b2 = GCD1b, r2,a Ú b,

1n + 121m + 12.

pnqmp2 … n,

C. The least common multiple (LCM) of two ormore natural numbers is the least positive mul-tiple that the numbers have in common.

D.E. If then a and b are relatively

prime.VI. Modular arithmetic

A. For any integers a and b, a is congruent to bmodulo m if, and only if, is a multiple ofm, where m is a positive integer greater than 1.

B. Two integers are congruent modulo m if, andonly if, their remainders upon division by m arethe same.

a - b

GCD1a, b2 = 1,GCD1a, b2 # LCM1a, b2 = ab.

Chapter Review

1. Find the additive inverse of each of the following:a. 3 b. c.d. e. f.g. h.

2. Perform each of the following operations:a. b.c. d.e. f.

3. For each of the following, find all integer values ofx (if there are any) that make the given equationtrue:a.b.c.d.e.f.

4. Use a pattern approach to explain why 5. In each of the following chip models, the encircled

chips are removed. Write the corresponding integerproblem with its solution.

1-221-32 = 6.-2x + 3x = x3x - 1 = -124-x , 0 = -10 , 1-x2 = 0-2x = 10-x + 3 = 0

1-25 , 521-32-40 , 1-52-31-5 + 52-31-22 + 2-2 - 1-52 + 51-2 + -82 + 3

-251-225-x - y-x + yx + y

-2 + 3-ac.d.e.f.g.

7. Factor each of the following expressions and thensimplify, if possible:a.b.c.d.e.f.

8. Classify each of the following as true or false (allletters represent integers). Justify your answers.a. always is positive.b. For all x and y, c. If then d. For all x and y,

9. Find a counterexample to disprove each of the fol-lowing properties on the set of integers:a. Commutativity for divisionb. Associativity for subtractionc. Closure for divisiond. Distributive property of division over subtraction

10. Solve each of the following for x, where x is aninteger:a.b.c.d.e.f. ƒ x ƒ = -x

ƒ -x ƒ = 3-x = x210x = 2992x = -2100x + 3 = -x - 17

1x - y22 = 1 y - x22.a 6 0.a 6 -b,ƒ x + y ƒ = ƒ x ƒ + ƒ y ƒ .

ƒ x ƒ

1x - y21x + 12 - 1x - y25 + 5x81y4 - 16x4x2 - 36x2 + xx - 3x

1-2 - x21-2 + x21-3 - x213 + x21-x23 + x31-x22 + x22x - 11 - x2

(a) (b)

6. Simplify each of the following expressions:a.b. 1-121x - y2

-1x

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g.h.

11. Write the first six terms of each of the sequenceswhose nth term isa.

b.

c.d.

12. In each part of problem 11, if a sequence is arith-metic, find its difference, and if it is geometric, findits ratio.

13. Classify each of the following as true or false:a.b.c.d. If a number is divisible by 4 and by 6, then it is

divisible by 24.e. If a number is not divisible by 12, then it is not

divisible by 3.14. Classify each of the following as true or false. If

false, show a counterexample.a. If and then b. If then and c. If and then d. If and , then .e. If and then

15. Test each of the following numbers for divisibilityby 2, 3, 4, 5, 6, 8, 9, and 11:a. 83,160b. 83,193

16. Assume that 10,007 is prime. Without actually di-viding 10,024 by 17, prove that 10,024 is not divisi-ble by 17.

17. Fill each blank with one digit to make each of thefollowing true (find all the possible answers):a.b.c.

18. A student claims that the sum of five consecutivepositive integers is always divisible by 5.a. Check the student’s claim for a few cases.b. Prove or disprove the student’s claim.

19. Determine whether each of the following numbersis prime or composite:a. 143 b. 223

20. How can you tell if a number is divisible by 24?Check 4152 for divisibility by 24.

21. Is the LCM of two numbers always greater than theGCD of the numbers? Justify your answer.

22. Explain how to find the LCM of three numberswith the help of the Euclidean algorithm.

23. To find if the number is prime,a student finds that the number equals 353. Shechecks that and 353 and without192 717�353

2 # 3 # 5 # 7 + 11 # 13

29 ƒ 87__424 ƒ 4_8566 ƒ 87_4

4�xy.4�y,4�xd ƒ yd ƒ xd ƒ 1x + y2d�b.d�a,d ƒ 1a + b2d�b.d�ad�1a + b2,

7�xy.7�y,7 ƒ x

4 ƒ 00 ƒ 48 ƒ 4

-2 - 3n1-22n

n2

31-12n + 14

1-12n

1x - 122 = 100ƒ x ƒ 7 3 further checking, claims that 353 is prime. Explain

why the student is correct.24. Find the GCD for each of the following:

a. 24 and 52b. 5767 and 4453

25. Find the LCM for each of the following:a. and b. 278 and 279

26. Construct a number that has exactly five positivedivisors. Explain your construction.

27. Find all the positive divisors of 144.28. Find the prime factorization of each of the following:

a. 172 b. 288c. 260 d. 111

29. Find the least positive number that is divisible byevery positive integer less than or equal to 10.

30. Candy bars priced at 50¢ each were not selling, sothe price was reduced. Then they all sold in one dayfor a total of $31.93. What was the reduced price ofeach candy bar?

31. Two bells ring at 8:00 A.M. For the remainder of theday, one bell rings every half hour and the other bellrings every 45 min. What time will it be when thebells ring together again?

32. If the GCD of two positive whole numbers is 1,what can you say about the LCM of the two num-bers? Explain your reasoning.

33. If there were to be 9 boys and 6 girls at a party andthe host wanted each to be given exactly the samenumber of candies that could be bought in packagescontaining 12 candies, what is the fewest number ofpackages that could be bought?

34. Jane and Ramon are running laps on a track. If theystart at the same time and place and go in the samedirection, with Jane running a lap in 5 min andRamon running a lap in 3 min, how long will it takefor them to be at the starting place at the same timeif they continue to run at these speeds?

35. June, an owner of a coffee stand, marked down theprice of a latte between 7:00 A.M. and 8:00 A.M.from $2.00 a cup. If she grossed $98.69 from thelatte sale and we know that she never sells a lattefor less than a dollar, how many lattes did she sellbetween 7:00 A.M. and 8:00 A.M.? Explain your rea-soning. (Note: )

36. Find the prime factorizations of each of the fol-lowing.a.b.c.d.e.f.

37. What are the possible remainders when a primenumber greater than 3 is divided by 12? Justify youranswer.

24 # 3 # 57 + 24 # 5623 # 32 + 24 # 33 # 784 # 63 # 26297434n610

71 ƒ 9869.

24 # 5 # 74 # 2923 # 52 # 73, 2 # 53 # 72 # 13

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Selected Bibliography 339

38. Prove the test for divisibility by 9 using a three-digitnumber n such that

39. The triplet 3, 5, 7 consists of consecutive odd inte-gers that are all prime. Give a convincing argumentthat this is the only triplet of consecutive oddintegers that are all prime. (Hint: use the division al-gorithm.)

40. The length of a week was probably inspired by theneed for market days and religious holidays. TheRomans, for example, once used an 8-day week.

n = a # 102 + b # 10 + c.Assuming April still had 30 days but was based onan 8-day week, if the first day of the month was onSunday and the extra day after Saturday was calledVenaday, on what day would the last day of themonth fall?

41. To measure angles of rotation (in degrees) that alight on a small island lighthouse sweeps, what modsystem would be used and why?

Selected Bibliography

Anthony, G., and M. Walshaw. “Zero A ‘None’ Num-ber?” Teaching Children Mathematics 11 (August2004): 38–42.

Bay, J. “Developing Number Sense on the NumberLine.” Mathematics Teaching in the Middle School 6(April 2001): 448–451.

Bennett, A., and L. Nelson. “Divisibility Tests: So Rightfor Discoveries.” Mathematics Teaching in the MiddleSchool 7 (April 2002): 460–464.

Bezuszka, S., and M. Kenney. “Even Perfect Numbers:(Update)2.” Mathematics Teacher 90 (November1997): 628–633.

Brown, E., and E. Jones. “Using Clock Arithmetic toTeach Algebra Concepts.” Mathematics Teaching inthe Middle School 11 (September 2005): 104–109.

Graviss, T., and J. Greaver. “Extending the NumberLine to Make Connections with Number Theory.”Mathematics Teacher 85 (September 1992): 418–420.

Gregg, J., and D. Gregg. “A Context for Integer Com-putation.” Mathematics Teaching in the Middle School13 (August 2007): 46–50.

Nurnberger-Haag, J. “Integers Made Easy: Just Walk ItOff,” Mathematics Teaching in the Middle School 13(September 2007): 118–121.

Peterson, J. “Fourteen Different Strategies for Multi-plication of Integers, or Why .”Arithmetic Teacher 19 (May 1972): 396–403.

1-121-12 = 1+12

Petrella, G. “Subtracting Integers: An Affective Les-son.” Mathematics Teaching in the Middle School 7(November 2001): 150–151.

Ponce, G. “It’s All in the Cards: Adding and Subtract-ing Integers.” Mathematics Teaching in the MiddleSchool 13 (August 2007): 10–17.

Reeves, A., and M. Beasley. “Advanced Paint by Num-bers.” Mathematics Teaching in the Middle School 12(April 2007): 447.

Robbins, C., and T. Adams. “Get Primed to the BasicBuilding Blocks of Numbers.” Mathematics Teachingin the Middle School 13 (September 2007): 122–127.

Schneider, S., and C. Thompson. “Incredible EquationsDevelop Incredible Number Sense.” Teaching Chil-dren Mathematics 7 (November 2000): 146–148,165–168.

Shultz, H. “The Postage-Stamp Problem, NumberTheory, and the Programmable Calculator.”Mathematics Teacher 92 (January 1999): 20–22.

Steinberg, R., D. Sleeman, D. Ktorza. “Algebra StudentsKnowledge of Equivalent Equations.” Journal ofResearch in Mathematics Education 22 (February 1990):112–121.

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