Multiplying and Dividing Fractions Chapter 2 Multiplying and Dividing Fractions

Factors and Prime Factorization 2.2 Factors and Prime Factorization

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. Objective A 3

Prime and Composite Numbers Prime Numbers A prime number is a natural number that has exactly two different factors 1 and itself. Composite Numbers A composite number is any natural number, other than 1, that is not prime. Objective A 4

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. Objective A Composite, it has more than two factors: 1, 7, 49. 5

Prime Factorization Prime Factorization The prime factorization of a number is the factorization in which all the factors are prime numbers. Every whole number greater than 1 has exactly one prime factorization. Objective A 6

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. Objective A 7

Divisibility Tests Objective A 8

Factor Trees Another way to find the prime factorization is to use a factor tree. Objective A 9

Examples 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. Objective A 10

Examples 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. Objective A 11