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Factors and Prime Factorization

Section 2.2

Slide 2Copyright © 2015, 2011, 2008 Pearson Education, Inc.

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.

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Examples

Find all the factors of each number.

a. 15

b. 7

c. 24

1, 3, 5, 15

1, 7

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

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Prime and Composite Numbers

Prime Numbers

A prime number is a natural number that has exactly two different factors 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, … .

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

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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.

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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.

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Example

Find the prime factorization of 30.

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.

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Example

Find the prime factorization of 36.

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.

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Divisibility Tests

Slide 10Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Example

Write 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

Slide 11Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Example

Find the prime factorization of 45.

Write 45 as the product of two numbers. Continue until all factors are prime.

45

9 • 5

3 • 3

The prime factorization of 45 is 3 · 3 · 5 or 32 · 5.

Slide 12Copyright © 2015, 2011, 2008 Pearson Education, Inc.

Example

Find the prime factorization of 72.

Write 72 as the product of two numbers. Continue until all factors are prime.

72

9 • 8

3 • 3 4 • 2

2 • 2 3 22 2 2 3 3 or 2 3