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

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Presentation transcript:

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

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.

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

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.

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.

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.

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

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

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

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.

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 The prime factorization of 45 is 3 · 3 · 5 or 3 2 · 5.

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