4 1 Factors and Prime Factorization Learn to

4 -1 Factors and Prime Factorization Learn to write prime factorizations of composite numbers.

4 -1 Factors and Prime Factorization Vocabulary factor prime factorization

4 -1 Factors and Prime Factorization Whole numbers that are multiplied to find a product are called factors of that product. A number is divisible by its factors. 2 3=6 Factors Product 6 ÷ 3 = 2 6 ÷ 2 = 3 6 is divisible by 3 and 2.

4 -1 Factors and Prime Factorization Helpful Hint When the pairs of factors begin to repeat, then you have found all of the factors of the number you are factoring.

4 -1 Factors and Prime Factorization Additional Example 1 A: Finding Factors List all of the factors of the number 16. A. 16 16 = 1 • 16 16 = 2 • 8 16 = 4 • 4 16 = 8 • 2 1 2 4 4 1 2 3 4 5 6 7 8 is a factor. is not a factor. and 2 have already been listed so stop here. 8 16 You can draw a diagram to illustrate the factor pairs. The factors of 16 are 1, 2, 4, 8, and 16.

4 -1 Factors and Prime Factorization Additional Example 1 B: Finding Factors List all of the factors of the number 19. B. 19 19 = 1 • 19 19 is not divisible by any other whole number. The factors of 19 are 1 and 19.

4 -1 Factors and Prime Factorization Check It Out: Example 1 A List all of the factors of the number 12. A. 12 12 12 = = 1 2 3 4 • • 12 6 4 3 1 is a factor. 2 is a factor. 3 is a factor. 4 and 3 have already been listed so stop here. 4 6 12 You can draw a diagram to illustrate the factor pairs. The factors of 12 are 1, 2, 3, 4, 6, and 12

4 -1 Factors and Prime Factorization Check It Out: Example 1 B List all of the factors of the number 11. B. 11 11 = 1 • 11 11 is not divisible by any other whole number. The factors of 11 are 1 and 11.

4 -1 Factors and Prime Factorization You can use factors to write a number in different ways. Factorization of 12 1 • 12 2 • 6 3 • 4 3 • 2 Notice that these factors are all prime. The prime factorization of a number is the number written as the product of its prime factors.

4 -1 Factors and Prime Factorization Helpful Hint You can use exponents to write prime factorizations. Remember that an exponent tells you how many times the base is a factor.

4 -1 Factors and Prime Factorization Additional Example 2 A: Writing Prime Factorizations Write the prime factorization of 24. Method 1: Use a factor tree. Choose any two factors of 24 to begin. Keep finding factors until each branch ends at a prime factor. 24 2 • 6 12 • 6 2 • 3 3 • 2 4 2 • 2 24 = 3 • 2 • 2 24 = 2 • 2 • 3 The prime factorization of 24 is 2 • 2 • 3, or 23 • 3.

4 -1 Factors and Prime Factorization Additional Example 2 B: Writing Prime Factorizations Write the prime factorization of 45. Method 2: Use a ladder diagram. Choose a prime factor of 45 to begin. Keep dividing by prime factors until the quotient is 1. 3 45 5 15 3 5 5 1 45 = 3 • 5 45 9 3 3 3 1 45 = 5 • 3 The prime factorization of 45 is 3 • 5 or 32 • 5.

4 -1 Factors and Prime Factorization In Example 2, notice that the prime factors may be written in a different order, but they are still the same factors. Except for changes in the order, there is only one way to write the prime factorization of a number.

4 -1 Factors and Prime Factorization Check It Out: Example 2 A Write the prime factorization of 28. Method 1: Use a factor tree. Choose any two factors of 28 to begin. Keep finding factors until each branch ends at a prime factor. 28 2 • 28 14 2 • 7 7 28 = 2 • 7 • 4 2 • 2 28 = 7 • 2 The prime factorization of 28 is 2 • 7, or 22 • 7.

4 -1 Factors and Prime Factorization Check It Out: Example 2 B Write the prime factorization of 36. Method 2: Use a ladder diagram. Choose a prime factor of 36 to begin. Keep dividing by prime factors until the quotient is 1. 3 36 3 12 2 3 12 3 6 2 36 4 2 3 1 36 = 3 • 2 • 3 2 2 1 36 = 3 • 2 • 2 The prime factorization of 36 is 3 • 2 • 3, or 32 • 23.