Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 2.2 Factors and Prime...

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

2.2

Factors and Prime Factorization

Martin-Gay, Basic Mathematics, 4e 22


Finding the Factors of Numbers

To perform many operations, it is necessary to be able to factor a number.

Since 7 · 9 = 63, both 7 and 9 are factors of 63, and 7 · 9 is

called a factorization of 63.



Definition Example

Factors: When numbers are multiplied to form a product, each number is called a factor.

The different factorizations of 6 are:

So the factors of 6 are: 1, 2, 3 and 6.

32 and 61

P 122

Practice Problems 1Find all the factors of each of the numbers.

24 c.

7 .b

15 a. 53 151 and 15 5, 3, 1, :isAnswer 71 7 1, :isAnswer

64 83 12,2 ,241 and

24 12, 8, 6, 4, 3, 2, 1, :isAnswer

Finding Factors of Numbers



Prime and Composite Numbers

Prime NumbersA prime number is a natural number that has exactly two different factors 1 and itself.

Composite NumbersA composite number is any natural number, other than 1, that is not prime.

pp 122-123



Examples

Determine whether each number is prime or composite. Explain your answers.

a. 16

b. 31

c. 49

Composite, it has more than two factors: 1, 2, 4, 8, 16.

Prime, its only factors are 1 and 31.

Composite, it has more than two factors: 1, 7, 49.



Definition Example

Prime Number: A natural number that has exactly two different factors, 1 and itself.

The first several prime numbers are:

2, 3, 5, 7, 11, 13, 17

Composite Number: If a natural number other than 1 is not a prime number, it is called a composite number.

The number 10 has more than two factors:

1, 2, 5, and 10

p 123

Practice Problems 2Determine whether each number is

Prime or

Composite.

39 29 18 13 21

21

13

18

29

39

Identifying Prime and Composite Numbers



Prime Factorization

Every whole number greater than 1 has exactly one prime factorization.

Prime FactorizationThe prime factorization of a number is the factorization in which all the factors are prime numbers.

p 123



Examples

Find the prime factorization of 63.

The first prime number 2 does not divide evenly, but 3 does.

Because 21 is not prime, we divide again.

The quotient 7 is prime, so we are finished. The prime factorization of 63 is 3 · 3 · 7.

213 63

73 21

3 63



Definition Example

Prime Factorization: The factorization in which all the factors are prime numbers.

The prime factorization for 84.

842

422

42

21

2137

because 42 is not a prime number we must divide it by a prime number.

because 21 is not a prime number we must divide it by a prime number.

because 7 is a prime number we can now write the prime factorization of 84.

.732 is 84for ion factorizat prime The 2

p 124

Finding Prime Factorizations



Factor Trees

Another way to find the prime factorization is to use a factor tree.



Definition Practice Problem 3Prime Factorization: The factorization in which all the factors are prime numbers.

p 125

28

2 14

2 7

72or 72 2 isAnswer 2 282

142

7




Examples


Write 30 as the product of two numbers. Continue until all factors are prime.

30

6 • 5

3 • 2 • 5

The prime factorization of 30 is 2 · 3 · 5.



Examples


Write 36 as the product of two numbers. Continue until all factors are prime.

36

9 • 4

3 • 3 2 • 2

The prime factorization of 36 is 3 · 3 · 2 · 2 or 32 · 22.



Definition Practice Problem 4


p 124

120

2 60

2 30

532or 5322 2 isAnswer 3

2 15

3 5

1202

602

302

153

5






p 125

756

6 126

2 63

732or 733322 isAnswer 32

3 21

3 7

2 3

7562

3782

1893

633

213

7






p 125

70

2 35

5 7

75 2 isAnswer 702

355

7




Definition Practice Problem 7a


p 126

30

2 15

5 3

53 2 isAnswer 302

155

3




Definition Practice Problem 7b


p 126

56

2 28

2 14

72or 722 2 isAnswer 3 562

282

142

7

2 7

Problem 7c: 72 and Problem 8: 117




DONE



Divisibility Tests

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 2.2 Factors and Prime...

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