1. Slide 1 Copyright © 2015, 2011, 2008 Pearson Education, Inc. Factors and Prime Factorization Section2.2

2. Slide 2 Copyright © 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.

3. Slide 3 Copyright © 2015, 2011, 2008 Pearson Education, Inc. 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

4. Slide 4 Copyright © 2015, 2011, 2008 Pearson Education, Inc. 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 Numbers A composite number is a natural number, other than 1, that is not prime.

5. Slide 5 Copyright © 2015, 2011, 2008 Pearson Education, Inc. 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.

6. Slide 6 Copyright © 2015, 2011, 2008 Pearson Education, Inc. Prime Factorization Every whole number greater than 1 has exactly one prime factorization. Prime Factorization The prime factorization of a number is the factorization in which all the factors are prime numbers.

7. Slide 7 Copyright © 2015, 2011, 2008 Pearson Education, Inc. 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.

8. Slide 8 Copyright © 2015, 2011, 2008 Pearson Education, Inc. 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 3 2 · 2 2.

9. Slide 9 Copyright © 2015, 2011, 2008 Pearson Education, Inc. Divisibility Tests

10. Slide 10 Copyright © 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.

11. Slide 11 Copyright © 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 3 2 · 5.

12. Slide 12 Copyright © 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