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Chapter 4

Number Theory

4A Introduction to Indices (pg. 55)

4B Square Roots and Cube Roots (pg. 57)

4C Index Laws (pg. 59)

4D Divisibility (pg. 61)

4E Factors and Multiples (pg. 62)

4F Prime Numbers (pg. 64)

4G Highest Common Factor and Lowest Common Multiple (pg. 66)

Solutions (pg. 68)

Written by

Benjamin Odgers

Mathematics Teacher

B Teaching / B Science

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4A Introduction to Indices https://youtu.be/yOwm3JLFucY

Question 1

Which expression below has a base number of 11 and an exponent of 4? Circle the correct solution.

(a) 11 × 4 (b) 411 (c) 114 (d) 4 ÷ 11

Example 1 https://youtu.be/RZEYBka4DjI

a) Write the following expressions in index form:

(i)

3 × 3 × 3 × 3 × 3 (ii) 5 × 7 × 5 × 5 × 5 × 7 × 2

b) Expand the following expressions (do not solve):

(i)

23 × 62 (ii) 73 × 31 × 44

Question 2

Write the following expressions in index form:

(a)

4 × 4 × 4 (b) 7 × 7 (c) 1 × 1 × 1 × 1 × 1

(d)

10 × 10 × 10 × 10 (e) 25 (f) 2 × 2 × 2 × 2 × 2 × 2

Question 3

Write the following expressions in index form:

(a)

2 × 2 × 2 × 6 × 6 (b) 5 × 3 × 5 × 3 × 3 × 3

(c)

4 × 4 × 7 × 4 × 7 × 7 × 4 (d) 9 × 7 × 7 × 8 × 9 × 9 × 8

(e)

1 × 10 × 1 × 1 × 13 × 10 (f) 17 × 11 × 11 × 5 × 11 × 5 × 11

Question 4

Expand the following expressions (do not solve):

(a)

53 (b) 25 (c) 114

(d)

72 × 43 (e) 123 × 63 (f) 45 × 152

(g)

201 × 95 (h) 142 × 73 × 8 (i) 54 × 31 × 171

53 = 5 × 5 × 5

_________

_________ or exponent or ________

Index Form

53

Expanded Form

5 × 5 × 5

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Example 2 https://youtu.be/laRTyMayo3s

Expand then solve the following, check your solution using a scientific calculator. The index button usually

looks like either one of the pictures shown.

(a)

26 (b) 24 × 33 (c) 53 − 34

Question 5

Expand then solve the following, check your solution using a scientific calculator:

(a)

25 (b) 104 (c) 23 × 15

(d)

32 × 52 (e) 42 + 33 (f) 43 − 52 × 2

Question 6

Solve the following using a scientific calculator.

(a)

57 (b) 86 − 57 (c) 75 × 123

(d)

(12 + 13)4 (e) (47 + 210) × 34 (f) (53 − 25)4

Question 7

Fill in the gaps below to make each statement true.

(a) 5 = 25 (b) 2 = 64 (c) 3 = 81 (d) 4 = 64

Question 8

Let’s say you have $1 in your bank account and double this amount every year.

a) How much money would you have after 20 years? Express your solution in dollars and also in index

form.

b) How many years would it take until you had over one billion dollars in your account?

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4B Square Roots and Cube Roots https://youtu.be/DnK--QqhOCA

Squaring Cubing Square Roots Cube Roots

82

8 squared

8 to the power of 2

53

5 cubed

5 to the power of 3

√9

√92

square root of 9

√643

cube root of 64

Example 1 https://youtu.be/IHvOVg-zaow

Solve the following without a calculator:

(a)

6 squared (b) √49 (c) √273

(d) square root of 16 (e) 4 cubed (f) cube root of 125

Question 1 Solve the following without a calculator:

(a)

92 (b) 3 cubed (c) √16

2

(d)

7 squared (e) 43 (f) 112

(g)

√64 (h) √8

3 (i) square root of 25

(j) cube root of 1000 (k) √121 (l) √643

Question 2 Solve the following without a calculator:

(a)

√4 (b) √1

3 (c) √81

2

(d)

√1 000 0003

(e) √144 (f) √2163

(g)

√5123

(h) √9002

(i) √400

(j) √169 (k) √10 0002

(l) √13313

Question 3 Solve the following without a calculator:

(a)

52 + √16 (b) √273

× 3 (c) 43 − 33

(d)

√62 (e) √3 × 5 + 1 (f) √32 + 42

(g)

√643

÷ √83

(h) √533 (i) √132 − 52

52 = × =

25 = ×

∴ √25 =

23 = × × =

8 = × ×

∴ √83

=

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Example 2 https://youtu.be/rMcpwDiMdYQ Use a calculator to solve the following:

(a)

142 − √1681 (b) √2197

3× 2.13 (c)

√122 + 72

6

Question 4 Use a calculator to solve the following:

(a)

√256 + 123 (b) 45 ÷ √33753

(c) √152 + 362

(d)

√

98 + 8.48

10

3

(e)

√122 + 162

4

(f)

(0.9)3 − √142

282

Example 3 https://youtu.be/wfFgkeL1R2s

a) What are square numbers?

b) Name all the square numbers between 1 and 20

c) Name two square numbers that have a difference of 56

Question 5

a) Name all the square numbers between 1 and 200 (note: there are only 14 of them)

b) Name two square numbers that have a sum of 146

c) Name two square numbers that have a difference of 57

d) Name three square numbers that have a sum of 134

Question 6

Give a value for x such that √√√𝑥 = 2.

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4C Index Laws

Index Law 1: Multiplication https://youtu.be/7aQQi_HmcF8

If you multiply terms with the same base, you can add the powers.

For example: 73 × 72 = 73+2 = 75

Example 1

Simplify the following, giving your solution in index form:

(a) 23 × 27 (b) 52 × 53 × 54 (c) 34 × 3

Question 1

Simplify the following, giving your solution in index form:

(a)

62 × 68 (b) 133 × 1311 (c) 93 × 95 × 910

(d) 25 × 2 (e) 102 × 109 × 10 (f) 72 × 7 × 77 × 71

Index Law 2: Division https://youtu.be/2pWJ10KXVvE

If you divide terms with the same base, you can subtract the powers.

For example: 45 ÷ 43 = 45−3 = 42 similarly 45

43 = 42

Example 2

Simplify the following, giving your solution in index form:

(a) 39 ÷ 34 (b) 65 ÷ 6 (c) 87

86

Question 2

Simplify the following, giving your solution in index form:

(a)

57 ÷ 53 (b) 1010 ÷ 106 (c) 94 ÷ 9

(d) 28

23

(e) 713

7

(f) 159

158

Index Law 3: Power of a Power https://youtu.be/4u3ZfCAG_kM

If you raise a power to another power, you multiply the powers.

For example: (92)3 = 92×3 = 96

Example 3

Simplify the following, giving your solution in index form:

(a) (53)4 (b) (115)1 (c) (83)12

Question 3

Simplify the following, giving your solution in index form:

(a)

(74)5 (b) (35)2 (c) (47)6

(d) (127)1 (e) (1610)3 (f) (131)13

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Index Law 4: Zero Index https://youtu.be/MNedrjtjcWI

If you raise a number to the power of 0 you get 1. Check this on your calculator.

For example: 130 = 1

Example 4

Solve the following without using a calculator:

(a)

70 (b) (13 + 11)0 (c) 13 + 110

(d) (−14)0 (e) −140 (f) 21 × 30 + 150

Question 4

Solve the following without using a calculator:

(a)

150 (b) 12090 (c) 50 + 17

(d)

(12 × 13)0 (e) 6 × 170 (f) 140 + 130 × 7

(g)

(−21)0 (h) −210 (i) (3 × 25)0

(j)

(4 + 3 × 2)0 − 210 (k) −(12 − 5 × 4)0 (l) (

3

4)

0

× (6

8)

0

Example 5 https://youtu.be/BMTeTKa67_8

Simplify the following, giving your solution in index form:

(a) 32 × (32)5 (b) (

610

67)

4

(c) 73 × 7 ÷ 72 × 50

Question 5

Simplify the following, giving your solution in index form:

(a)

(24)3 ÷ 28 (b) 63 × (65)3 (c) 95 ÷ 92 × 94

(d)

(35 × 32)3 (e) (56)4 × 5 (f) (116)2 ÷ (113)2

(g)

(49

44)

2

(h)

(136

133)

3

× (135

134)

2

(i) (154)4 ÷ (153)3 × 90

Question 6

Solve the following without using a calculator:

(a)

60 × (35)2 ÷ 38 (b) (133)4 ÷ (136)2 (c) 70 + 33 × (264)0

(d)

63 × 67

68÷ 3

(e) 117

115× 11 ÷

119

118× 2

(f) (

88

84)

4

÷ (810

85)

3

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4D Divisibility https://youtu.be/0BV0tRZyIj0

A number is divisible by another number if after the division has taken place there is no remainder.

There are techniques we can use to check the divisibility of numbers. The table below shows you some of

these techniques.

Number How do we know what numbers are divisible by this?

1 All numbers are divisible by 1.

2 Any number where the last digit is a 0, 2, 4, 6 or 8.

3 Any number where the sum of the digits is divisible by 3.

4 Any number where the last two digits are divisible by 4.

5 Any number where the last digit is a 0 or a 5.

6 Any number that is divisible by both 2 and 3.

7 There is no easy way to do check the divisibility of 7.

8 Any number where the last 3 digits are divisible by 8.

9 Any number where the sum of the digits is divisible by 9.

10 Any number that has a last digit of 0.

Example 1 https://youtu.be/SAhS_k0NEbc

Find out if the following numbers are divisible by referring to the table above. Check your answer using a

calculator.

(a)

312 345 ÷ 2 (b) 97805 ÷ 5 (c) 21081 ÷ 3

(d) 23 412 ÷ 8 (e) 12 096 ÷ 9 (f) 3012 ÷ 6

Question 1

Find out if the following numbers are divisible by referring to the table above. Check your answer using a

calculator (write yes or no next to each question below).

(a)

3 184 ÷ 2 (b) 29 603 ÷ 10 (c) 724 ÷ 1

(d)

67 080 ÷ 5 (e) 59 031 ÷ 9 (f) 614 ÷ 4

(g)

130 503 ÷ 2 (h) 6 102 ÷ 3 (i) 5106 ÷ 6

(j)

24 950 ÷ 10 (k) 17 083 ÷ 9 (l) 21 095 ÷ 3

(m)

3 024 524 ÷ 4 (n) 3 795 ÷ 5 (o) 121 ÷ 6

(p)

91 ÷ 7 (q) 12 120 ÷ 8 (r) 17 232 ÷ 3

(s)

111 ÷ 3 (t) 120 987 ÷ 9 (u) 4 064 ÷ 8

(v) 6 780 ÷ 4 (w) 123 456 ÷ 6 (x) 111111111 ÷ 9

Question 2

A business man has $1205 in bonuses that he would like to share equally among 9 employees.

a) Is it possible for him to do this if he plans to pay each employee in whole dollar amounts (no cents)?

b) What is the minimum amount of money he can take away from the bonus so that every employee

gets whole dollar amounts?

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4E Factors and Multiples https://youtu.be/GfeX9Yrza2A

Example 1 https://youtu.be/xqwRk9F94aw

Write down all the factors for each number below.

(a)

9 (b) 24

Question 1

Fill in the gaps by writing down the factors for each number below.

(a)

12 1, ___, 3, 4, ___, 12

(b)

35 ___, 5, ___, 35

(c) 36 1, 2, ___, 4, ___, ___, 12, 18, ___

Question 2

Write down all the factors for each number below.

(a)

8 (b) 14 (c) 18

(d)

25 (e) 32 (f) 30

(g)

13 (h) 48 (i) 63

Example 2 https://youtu.be/tdBCM-sQgoo

Write down the first five multiples for each number below.

(a)

5 (b) 13

Question 3

Write down the first five multiples for each number below.

(a)

3 (b) 9 (c) 11

(d)

7 (e) 15 (f) 40

(g)

31 (h) 17 (i) 29

Question 4

Circle the numbers below that are multiples of 14.

28, 7, 42, 1, 3, 14, 72, 10, 70, 140, 0, 126, 98, 202

What is a factor?

What is a multiple?

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Question 5

Joanne goes to yoga classes every 3rd day and Leila goes to yoga classes every 7th day. How many days need

to pass for them both to attend yoga on the same day?

Question 6

William and Angela are riding around a bicycle circuit. Both cyclists start at the same time. William takes 8

minutes to complete a circuit and Angela takes 6 minutes to complete a circuit. How long will it take for

Angela to overtake (or lap) William?

Question 7

Buoy 1 flashes a light every 16 seconds and Buoy 2 flashes a light every 20 seconds. How long will it take

until both Buoys flash their lights at the same time?

Question 8

A marching band has 48 members that needs to be in a rectangular formation as

it marches down a street (see diagram to the right). Because of the width of the

street, the marching band can have a maximum width of 6 people. How many

different formations can the marching band make?

Question 9

There are 360 degrees in a full revolution. Many people have argued that it would be better to change a full

revolution to 400 degrees so that right angles would be 100 degrees.

a) What are your thoughts on this?

b) How many factors does the number 360 have?

c) How many factors does the number 400 have?

d) The reason we have kept a full revolution at 360 degrees is because 360 has many more factors than

400. Why do you think this is important?

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4F Prime Numbers https://youtu.be/uW7Km01ZEyw

Question 1

State whether the following numbers are prime (P), composite (C) or neither (N):

(a)

25 (b) 2 (c) 13 (d) 15

(e)

12 (f) 23 (g) 9 (h) 1

(i)

27 (j) 78 (k) 31 (l) 49

(m) 61 (n) 1084 (o) 51 (p) 89

Example 1 https://youtu.be/FyYQyE9_bNQ

Express the following numbers as a product of their prime factors:

(a)

90 (b) 3960

Question 2

Express the following numbers as a product of their prime factors:

(a)

30 (b) 40 (c) 63 (d) 150

What is the difference between a prime number and a composite number?

Is the number 1 prime? Why/why not?

List all the prime numbers from 1 to 20:

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Question 2 continued

Express the following numbers as a product of their prime factors:

(e)

165 (f) 245 (g) 990 (h) 7500

(i)

1400 (j) 726 (k) 612 (l) 209

Question 3

List all the composite numbers between 40 and 50.

Question 4

An “emirp” (prime spelled backwards) is a prime number that makes another prime number when the digits

are reversed. For example, 13 when reversed makes 31 (both 13 and 31 are prime). What other 2-digit

numbers can be referred to as an emirp?

Question 5

Why are all even numbers other than 2 composite?

Question 6

A semiprime is not a prime number but comes from the product of two primes. For example, 15 is

semiprime since it comes from the product of 5 and 3 (both of which are prime). Find the two primes for

each semiprime below:

(a) 33 (b) 365 (c) 111

Question 7

Twin primes are a pair of prime numbers that have a difference of 2. For example, 3 and 5 are twin primes

since 5 − 3 = 2. There are 8 sets of twin primes between 1 and 100. What are they?

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4G Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

Example 1 https://youtu.be/OCn0mjABDL8

Find the highest common factor (HCF) for each pair of numbers:

(a)

30 and 45 (b) 5 and 14

Question 1

Find the highest common factor (HCF) for each pair of numbers:

(a)

8 and 12 (b) 10 and 14 (c) 18 and 24

(d)

7 and 14 (e) 16 and 20 (f) 3 and 11

(g)

35 and 95 (h) 28 and 49 (i) 9 and 54

Example 2 https://youtu.be/ShP1FTHB6qY

Find the lowest common multiple (LCM) for each pair of numbers:

(a)

4 and 10 (b) 3 and 9

Question 2

Find the lowest common multiple (LCM) for each pair of numbers:

(a)

4 and 6 (b) 6 and 15 (c) 3 and 5

(d)

6 and 18 (e) 9 and 12 (f) 7 and 11

(g)

15 and 25 (h) 14 and 42 (i) 6 and 32

Give a definition for highest common factor (HCF):

Give a definition for lowest common multiple (LCM):

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Example 3 https://youtu.be/cly29qoiTnk

Find the highest common factor (HCF) for each pair of numbers using the prime factor method:

(a)

36 and 48 (b) 42 and 60

Question 3

Find the highest common factor (HCF) for each pair of numbers using the prime factor method:

(a)

90 and 120 (b) 60 and 84 (c) 56 and 72

(d)

85 and 102 (e) 132 and 231 (f) 350 and 735

Example 4 https://youtu.be/TifmpKaEk6U

Find the lowest common multiple (LCM) for each pair of numbers using the prime factor method:

(a)

24 and 36 (b) 42 and 60

Question 4

Find the lowest common multiple (LCM) for each pair of numbers using the prime factor method:

(a)

12 and 18 (b) 15 and 50 (c) 28 and 126

(d)

40 and 84 (e) 21 and 110 (f) 35 and 231

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Solutions

Chapter 4A

Q1 – (c)

Q2 – (a) 43 (b) 72 (c) 15 (d) 104 (e) 251 (f) 26

Q3 – (a) 23 × 62 (b) 34 × 52 (c) 44 × 73 (d) 72 × 82 × 93

(e) 13 × 102 × 131 (f) 52 × 114 × 171

Q4 – (a) 5 × 5 × 5 (b) 2 × 2 × 2 × 2 × 2 (c) 11 × 11 × 11 × 11

(d) 7 × 7 × 4 × 4 × 4 (e) 12 × 12 × 12 × 6 × 6 × 6

(f) 4 × 4 × 4 × 4 × 4 × 15 × 15 (g) 20 × 9 × 9 × 9 × 9 × 9

(h) 14 × 14 × 7 × 7 × 7 × 8 (i) 5 × 5 × 5 × 5 × 31 × 17

Q5 – (a) 32 (b) 10 000 (c) 8 (d) 225 (e) 43 (f) 14

Q6 – (a) 78 125 (b) 184 019 (c) 29 042 496 (d) 390 625 (e) 1 410 048

(f) 74 805 201

Q7 – (a) 2 (b) 6 (c) 4 (d) 3

Q8 – (a) 220 = $1 048 576 (b) 230 = $1 073 741 824 (30 years)

Chapter 4B

Q1 – (a) 81 (b) 27 (c) 4 (d) 49 (e) 64 (f) 121 (g) 8 (h) 2 (i) 5 (j) 10 (k) 11

(l) 4

Q2 – (a) 2 (b) 1 (c) 9 (d) 100 (e) 12 (f) 6 (g) 8 (h) 30 (i) 20 (j) 13 (k) 100

(l) 11

Q3 – (a) 29 (b) 9 (c) 37 (d) 6 (e) 4 (f) 5 (g) 2 (h) 5 (i) 12

Q4 – (a) 1744 (b) 3 (c) 39 (d) 2.2 (e) 10 (f) 0.229

Q5 – (a) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196

(b) 121 and 25 (c) 121 and 64 (d) 81 + 49 + 4

Q6 – 256

Chapter 4C

Q1 – (a) 610 (b) 1314 (c) 918 (d) 26 (e) 1012 (f) 711

Q2 – (a) 54 (b) 104 (c) 93 (d) 25 (e) 712 (f) 15 or 151

Q3 – (a) 720 (b) 310 (c) 442 (d) 127 (e) 1630 (f) 1313

Q4 – (a) 1 (b) 1 (c) 18 (d) 1 (e) 6 (f) 8 (g) 1 (h) −1 (i) 1 (j) 0 (k) −1 (l) 1

Q5 – (a) 24 (b) 618 (c) 97 (d) 321 (e) 525 (f) 116 (g) 410 (h) 1311 (i) 157

Q6 – (a) 𝟗 (b) 1 (c) 28 (d) 12 (e) 242 (f) 8

Chapter 4D

Q1 – (a) Y (b) N (c) Y (d) Y (e) Y (f) N (g) N (h) Y (i) Y (j) Y (k) N (l) N (m) Y (n) Y (o) N (p) Y (q) Y (r) Y (s) Y (t) Y (u) Y (v) Y (w) Y

(x) Y Q2 – (a) No (b) $8

Chapter 4E

Q1 – (a) 1,2,3,4,6,12 (b) 1,5,7,35 (c) 1,2,3,4,6,9,12,18,36

Q2 – (a) 1,2,4,8 (b) 1,2,7,14 (c) 1,2,3,6,9,18 (d) 1,5,25 (e) 1,2,4,8,16,32

(f) 1,2,3,5,6,10,15,30 (g) 1,13 (h) 1,2,3,4,6,8,12,16,24,48

(i) 1,3,7,9,21,63

Q3 – (a) 3,6,9,12,15 (b) 9,18,27,36,45 (c) 11,22,33,44,55 (d) 7,14,21,28,35

(e) 15,30,45,60,75 (f) 40,80,120,160,200 (g) 31,62,93,124,155

(h) 17,34,51,68,85 (i) 26,58,87,116,145

Q4 – 0,28,42,14,70,140,126,98 Q5 – 21st day

Q6 – 24 minutes

Q7 – 80 seconds

Q8 – 5 formations (6 × 8, 4 × 12, 3 × 16, 2 × 24, 1 × 48)

Q9 – (a) any response possible (b) 24 factors (1,2,3,4,5,6,8,9,10,12,15,18,

20,24,30,36,40,45,60,72,90,120,180,360) (c) 15 factors (1,2,4,5,8,10,

16,20,25,40,50,80,100,200,400) (d) It means that you can cut a circle into several equal sized pieces (similar to pizza slices) and each

piece is more likely to have an angle that is a whole number.

Chapter 4F

Q1 – (a) C (b) P (c) P (d) C (e) C (f) P (g) C (h) N (i) C (j) C (k) P (l) C

(m) P (n) C (o) C (p) P

Q2 – (a) 2 × 3 × 5 (b) 23 × 5 (c) 32 × 7 (d) 2 × 3 × 52 (e) 3 × 5 × 11

(f) 5 × 72 (g) 2 × 32 × 5 × 11 (h) 22 × 3 × 54 (i) 23 × 52 × 7

(j) 2 × 3 × 112 (k) 22 × 32 × 17 (l) 11 × 19

Q3 – 40,42,44,45,46,48,49,50

Q4 – (17,71), (37,73), (79,97)

Q5 – Because they are all divisible by 2. Q6 – (a) 3,11 (b) 5,73 (c) 3,37

Q7 – (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73)

Chapter 4G

Q1 – (a) 4 (b) 2 (c) 6 (d) 7 (e) 4 (f) 1 (g) 5 (h) 7 (i) 9

Q2 – (a) 12 (b) 30 (c) 15 (d) 18 (e) 36 (f) 77 (g) 75 (h) 42 (i) 96

Q3 – (a) 30 (b) 12 (c) 8 (d) 17 (e) 33 (f) 35 Q4 – (a) 36 (b) 150 (c) 252 (d) 840 (e) 2310 (f) 1155