Prime Numbers

http://www.lahc.edu/math/frankma.htm

Prime NumbersIt we are to divide 6 pieces of candy into bag(s) so each bag has the same amount, we may do it in the following manner.

If we have 3 candies, we may bag them in the obvious manner

or

or

or

Besides dividing them in the “obvious” ways: put all 6 candies into one bag, or put one candy into 6 separate bags,

we may also divide the6 pieces into smaller bags as:

but we can’t bag them into other smaller bags evenly.

Because 3 can’t be divided into smaller groups, except as singles, we say that “3 is prime.”

On the other hand 6 = 2 x 3, i.e. 6 can be broken up into smaller groups, therefore 6 is not prime.

2 x 3 3 x 2

Let’s define these important relations precisely. Prime Numbers

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

Prime Numbers

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

Prime Numbers

In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6.

Prime Numbers

In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc.

Prime Numbers

In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc.

Prime Numbers

In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible?(These numbers are the factors of 12.)

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc..

Prime Numbers

In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible?(These numbers are the factors of 12.)

We may have 1 piece/per bag,

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc.

Prime Numbers

In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible?(These numbers are the factors of 12.)

We may have 1 piece/per bag, 2 pieces/bag,

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc.

Prime Numbers

In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible?(These numbers are the factors of 12.)

We may have 1 piece/per bag, 2 pieces/bag, 3 pieces/ bag,

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc.

Prime Numbers

In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible?(These numbers are the factors of 12.)

We may have 1 piece/per bag, 2 pieces/bag, 3 pieces/ bag, 4 pieces/bag, 6 pieces/bag, or 12 pieces/bag.

Let’s define these important relations precisely. If A, B and C are non-zero numbers such that A x B = C, then we say that “A (or B) is a factor of C”, or that “C is a multiple of A (or B)”.

For example, 6 may be dividedly 1, 2, 3 and 6 evenly, hence 1, 2, 3, and 6 are factors of 6, or 6 is a multiple of 1, 2, 3 and 6. Note that the multiples of 6 are 6(= 1x6) , 12(= 2x6) , 18(= 3x6) etc.

Prime Numbers

In other words, A is factor of C, or that C is a multiple of A, if C may be divided by A evenly.

Example A. a. 12 pieces of candy are to be put into one or more bags evenly. How many piece(s) of candy in each bag are possible?(These numbers are the factors of 12.)

We may have 1 piece/per bag, 2 pieces/bag, 3 pieces/ bag, 4 pieces/bag, 6 pieces/bag, or 12 pieces/bag. So the factors of 12 are 1, 2, 3, 4, 6, and 12.

Prime Numbersb. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.)

Prime Numbersb. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces.

Prime Numbersb. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, …

A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself.

Prime Numbersb. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, …

A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself.The following are the first few prime numbers: 2, 3, 5, 7, 11,13,..;

Prime Numbersb. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, …

A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself.The following are the first few prime numbers: 2, 3, 5, 7, 11,13,..;

Prime Numbers

but 4 = 2 x 2, 6 = 2 x 3, 8 = 2 x 2 x 2, 9 = 3 x 3, are not prime.

b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, …

A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself.The following are the first few prime numbers: 2, 3, 5, 7, 11,13,..;

Prime Numbers

but 4 = 2 x 2, 6 = 2 x 3, 8 = 2 x 2 x 2, 9 = 3 x 3, are not prime.

b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, …

The problem of determining easily which numbers are prime and which are not, is a major problem in mathematics.

A number C is a prime number if C only has 1 and itself as factors, i.e. C can be divided evenly only by 1 and itself.The following are the first few prime numbers: 2, 3, 5, 7, 11,13,..;

Prime Numbers

but 4 = 2 x 2, 6 = 2 x 3, 8 = 2 x 2 x 2, 9 = 3 x 3, are not prime.

b. A box contains bag(s) of candy and each bag has 12 pieces, how many candies in the box are possible? (These numbers are the multiples of 12.) There might be 1 bag or 2 bags or 3 bags, etc.. each with 12 pieces. So the box may have 12 =1x12, 24 =2x12, 36 =3x12, 48 =4x12, 60, ..., pieces, or that the multiples of 12 are 12, 24, 36, 48, …

http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

The link below has a good animation of one method of trying to determine which numbers are prime and which are not.

The problem of determining easily which numbers are prime and which are not, is a major problem in mathematics.

Exponents and Prime FactoringBelow we will write “A x B” as “A*B” for shorter notation.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Below we will write “A x B” as “A*B” for shorter notation.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

Below we will write “A x B” as “A*B” for shorter notation.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on. We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

Below we will write “A x B” as “A*B” for shorter notation.

N is called the exponent, or the power and x is called the base.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

Below we will write “A x B” as “A*B” for shorter notation.

==

N is called the exponent, or the power and x is called the base.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

Below we will write “A x B” as “A*B” for shorter notation.

= 3*3 = 9 =

N is called the exponent, or the power and x is called the base.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

Below we will write “A x B” as “A*B” for shorter notation.

= 3*3 = 9 =

(note that 32 is not 3*2 = 6)

N is called the exponent, or the power and x is called the base.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

Below we will write “A x B” as “A*B” for shorter notation.

= 3*3 = 9 = 4*4*4 = 64

(note that 32 is not 3*2 = 6)

N is called the exponent, or the power and x is called the base.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

Below we will write “A x B” as “A*B” for shorter notation.

= 3*3 = 9 = 4*4*4 = 64

(note that 32 is not 3*2 = 6)(note that 43 is not 4*3 = 12)

N is called the exponent, or the power and x is called the base.

The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

Below we will write “A x B” as “A*B” for shorter notation.

= 3*3 = 9 = 4*4*4 = 64

(note that 32 is not 3*2 = 6)(note that 43 is not 4*3 = 12)

N is called the exponent, or the power and x is called the base.

The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3.

Below we will write “A x B” as “A*B” for shorter notation.

= 3*3 = 9 = 4*4*4 = 64

(note that 32 is not 3*2 = 6)(note that 43 is not 4*3 = 12)

N is called the exponent, or the power and x is called the base.

The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number.

A factorization is complete if all the factors that appear in the multiplication are prime numbers.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3.

Below we will write “A x B” as “A*B” for shorter notation.

= 3*3 = 9 = 4*4*4 = 64

(note that 32 is not 3*2 = 6)(note that 43 is not 4*3 = 12)

N is called the exponent, or the power and x is called the base.

The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number.

A factorization is complete if all the factors that appear in the multiplication are prime numbers.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3.

Below we will write “A x B” as “A*B” for shorter notation.

= 3*3 = 9 = 4*4*4 = 64

(note that 32 is not 3*2 = 6)(note that 43 is not 4*3 = 12)

So 12 = 2*2*3 is factored completely because 2 and 3 are prime,

N is called the exponent, or the power and x is called the base.

The phrase to factor a number C means to write C as a product, i.e. to write C as # * # *** # where # might be any number.

A factorization is complete if all the factors that appear in the multiplication are prime numbers.

Exponents and Prime Factoring

Recall that to simplify the notation for repetitive multiplication, we write 2*2 as 22, 2*2*2 as 23, 2*2*2*2 as 24 and so on.

Example B. Calculate. 32 43

We write x*x*x…*x as xN where N is the number of times that x is multiplied to itself.

For example 12 can be factored as 12 = 2*6 = 3*4 = 2*2*3.

Below we will write “A x B” as “A*B” for shorter notation.

= 3*3 = 9 = 4*4*4 = 64

(note that 32 is not 3*2 = 6)(note that 43 is not 4*3 = 12)

So 12 = 2*2*3 is factored completely because 2 and 3 are prime, but 12 = 2* 6 is not factored completely because 6 is not a prime.

N is called the exponent, or the power and x is called the base.

Numbers that are factored completely may be written using the exponential notation.

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.Each number has a unique form when its completely factored into prime factors and arranged in order.

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format.

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format)

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format)

144

12 12

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format)

144

12 12

3 4 3 4

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format)

144

12 12

3 4 3 4

2 22 2

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format)

144

12 12

3 4 3 4

2 22 2

Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 32 * 24.

Exponents

Numbers that are factored completely may be written using the exponential notation.Hence, factored completely, 12 = 2*2*3 = 22

*3,200 = 8*25 = 2*2*2*5*5 = 23

* 52.Each number has a unique form when its completely factored into prime factors and arranged in order. This is analogous to chemistry where each chemical has a unique chemical composition of basic elements such as H20 for water.

Larger numbers may be factored using a vertical format. Example C. Factor 144 completely. (Vertical format)

144

12 12

3 4 3 4

2 22 2

Gather all the prime numbers at the end of the branches we have 144 = 3*3*2*2*2*2 = 32 * 24.Note that we obtain the same answer regardless how we factor at each step.

Exponents

Basic LawsWe summarize the basic laws of + and * operations again.

Basic Laws

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplication

We summarize the basic laws of + and * operations again.

Basic Laws

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a

We summarize the basic laws of + and * operations again.

Basic Laws

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12.

We summarize the basic laws of + and * operations again.

( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c )

Basic Laws

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12.

We summarize the basic laws of + and * operations again.

( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c )

Basic Laws

For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6.

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12.

We summarize the basic laws of + and * operations again.

( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c )

Basic Laws

For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6.

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12.

We summarize the basic laws of + and * operations again.

These laws allow us to + or * numbers in any order we wish.

( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c )

Basic Laws

For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6.

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12.

We summarize the basic laws of + and * operations again.

These laws allow us to + or * numbers in any order we wish.

Example D. Calculate.

14 + 3 + 16 + 8 + 35 + 15

( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c )

Basic Laws

For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6.

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12.

We summarize the basic laws of + and * operations again.

These laws allow us to + or * numbers in any order we wish.

Example D. Calculate.

14 + 3 + 16 + 8 + 35 + 15 = 30

( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c )

Basic Laws

For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6.

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12.

We summarize the basic laws of + and * operations again.

These laws allow us to + or * numbers in any order we wish.

Example D. Calculate.

14 + 3 + 16 + 8 + 35 + 15 = 30 + 50

( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c )

Basic Laws

For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6.

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12.

We summarize the basic laws of + and * operations again.

These laws allow us to + or * numbers in any order we wish.

Example D. Calculate.

14 + 3 + 16 + 8 + 35 + 15 = 30 + 50+ 11

( a + b ) + c = a + ( b + c ) ( a * b ) * c = a * ( b * c )

Basic Laws

For example, (1 + 2) + 3 = 3 + 3 = 6, is the same as 1 + (2 + 3) = 1 + 5 = 6.

The Associative Law for Addition and Multiplication

The Commutative Law for Addition and Multiplicationa + b = b + a a * b = b * a For example, 3 + 4 = 4 + 3 = 7 and 4*3 = 3*4 = 12.

We summarize the basic laws of + and * operations again.

These laws allow us to + or * numbers in any order we wish.

Example D. Calculate.

14 + 3 + 16 + 8 + 35 + 15 = 91 = 30 + 50+ 11

Basic LawsHowever, subtraction and division are not commutative nor associative.

Basic Laws

For example, 2 1 1 2,

However, subtraction and division are not commutative nor associative.

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2.

However, subtraction and division are not commutative nor associative.

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

However, subtraction and division are not commutative nor associative.

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

However, subtraction and division are not commutative nor associative.

Distributive Law: a*(b ± c) = a*b ± a*c

.

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

However, subtraction and division are not commutative nor associative.

Distributive Law: a*(b ± c) = a*b ± a*c

. Example E. Do 5*( 3 + 4 ) two different ways and show the results are the same.

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

However, subtraction and division are not commutative nor associative.

Distributive Law: a*(b ± c) = a*b ± a*c

. Example E. Do 5*( 3 + 4 ) two different ways and show the results are the same.

Do the ( ) first:

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

However, subtraction and division are not commutative nor associative.

Distributive Law: a*(b ± c) = a*b ± a*c

. Example E. Do 5*( 3 + 4 ) two different ways and show the results are the same.

Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35,

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

However, subtraction and division are not commutative nor associative.

Distributive Law: a*(b ± c) = a*b ± a*c

. Example E. Do 5*( 3 + 4 ) two different ways and show the results are the same.

Distribute the 5 first:

Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35,

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

However, subtraction and division are not commutative nor associative.

Distributive Law: a*(b ± c) = a*b ± a*c

. Example E. Do 5*( 3 + 4 ) two different ways and show the results are the same.

Distribute the 5 first:

Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35,

5*( 3 + 4 ) = 5*3 + 5*4

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

However, subtraction and division are not commutative nor associative.

Distributive Law: a*(b ± c) = a*b ± a*c

. Example E. Do 5*( 3 + 4 ) two different ways and show the results are the same.

Distribute the 5 first:

Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35,

5*( 3 + 4 ) = 5*3 + 5*4 = 15 + 20 = 35

Basic Laws

For example, 2 1 1 2, and that ( 3 – 2 ) – 1 = 0 is different from 3 – ( 2 – 1 ) = 2. Thus when subtracting or dividing, we have to determine who is in the front and who is in the back.

However, subtraction and division are not commutative nor associative.

Distributive Law: a*(b ± c) = a*b ± a*c

. Example E. Do 5*( 3 + 4 ) two different ways and show the results are the same.

Distribute the 5 first:

Do the ( ) first: 5*( 3 + 4 ) = 5*7 = 35,

5*( 3 + 4 ) = 5*3 + 5*4 = 15 + 20 = 35

the outcomes are the same as stated in the distributive law.

19. List the all the factors and the first 4 multiples of the following numbers. 6, 9, 10, 15, 16, 24, 30, 36, 42, 56, 60.

B. Calculate.1. 33 2. 42 3. 52 4. 53 5. 62 6. 63 7. 72

8. 82 9. 92 10. 102 11. 103 12. 104 13. 105

14. 1002 15. 1003 16. 1004 17. 112 18. 122

20. Factor completely and arrange the factors from smallest to the largest in the exponential notation: 4, 8, 12, 16, 18, 24, 27, 32, 36, 45, 48, 56, 60, 63, 72, 75, 81, 120.

21. 3 * 5 * 4 * 2 22. 6 * 5 * 4 * 323. 6 * 15 * 3 * 2

24. 7 * 5 * 5 * 4 25. 6 * 7 * 4 * 3 26. 9 * 3 * 4 * 4

27. 2 * 25 * 3 * 4 * 2 28. 3 * 2 * 3 * 3 * 2 * 4

29. 3 * 5 * 2 * 5 * 2 * 430. 4 * 2 * 3 * 15 * 8 * 4©

31. 24 32. 25 33. 26 34. 27 35. 28

36. 29 37. 210 38. 34 39. 35 40. 36

C. Multiply in two ways to find the correct answer.

© F. Ma

Exercise A. Do the following problems two ways. * Add the following by summing the multiples of 10 first. * Add by adding in the order.to find the correct answer. 1. 3 + 5 + 7 2. 8 + 6 + 2 3. 1 + 8 + 9 4. 3 + 5 + 15 5. 9 + 14 + 6 6. 22 + 5 + 8 7. 16 + 5 + 4 + 3 8. 4 + 13 + 5 + 79. 19 + 7 + 1 + 3 10. 4 + 5 + 17 + 311. 23 + 5 + 17 + 3 12. 22 + 5 + 13 + 2813. 35 + 6 + 15 + 7 + 14 14. 42 + 5 + 18 + 1215. 21 + 16 + 19 + 7 + 44 16. 53 + 5 + 18 + 27 + 2217. 155 + 16 + 25 + 7 + 344 18. 428 + 3 + 32 + 227 + 22Hence to add a list numbers, it's easier to add the ones that add to multiples of 10 first.Example D.

14 + 3 + 16 + 8 + 35 + 15 = 30 + 11 + 50 = 91