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Learn to write prime factorizations of composite numbers. Understand factors, prime factorization, and how to list all factors of a number. Discover helpful hints and examples involving factor pairs and prime factorizations.
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Vocabulary factor 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 6 3 6 2 3 = ÷ = 6 is divisible by 3 and 2. 3 6 ÷ 2 = Factors Product
Helpful Hint When the pairs of factors begin to repeat, then you have found all of the factors of the number you are factoring.
Additional Example 1A: Finding Factors List all of the factors of the number 16. A. 16 16 = 1 • 16 1 is a factor. 16 = 2 • 8 2 is a factor. 3 is not a factor. 16 = 4 • 4 4 is a factor. 5 is not a factor. 6 is not a factor. 7 is not a factor. 16 = 8 • 2 8 and 2 have already been listed so stop here. 1 2 4 4 8 16 You can draw a diagram to illustrate the factor pairs. The factors of 16 are 1, 2, 4, 8, and 16.
Additional Example 1B: 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.
Check It Out: Example 1A List all of the factors of the number 12. A. 12 12 = 1 • 12 1 is a factor. 12 = 2 • 6 2 is a factor. 12 = 3 • 4 3 is a factor. 12 = 4 • 3 4 and 3 have already been listed so stop here. 1 2 3 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
Check It Out: Example 1B 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.
Factorization of 12 You can use factors to write a number in different ways. Notice that these factors are all prime. 1 • 12 2 • 6 3 • 4 3 • 2 • 2 The prime factorization of a number is the number written as the product of its prime factors.
Helpful Hint You can use exponents to write prime factorizations. Remember that an exponent tells you how many times the base is a factor.
Additional Example 2A: 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 24 • 4 • 6 2 12 • • • 2 2 6 3 2 2 • 2 3 24 = 3 • 2 • 2 • 2 24 = 2 • 2 • 2 • 3 The prime factorization of 24 is 2 • 2 • 2 • 3, or 23 • 3.
Additional Example 2B: 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 45 15 9 3 3 5 3 5 3 1 1 45 = 3 • 3 • 5 45 = 5 • 3 • 3 The prime factorization of 45 is 3 • 3 • 5 or 32• 5 .
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.
Check It Out: Example 2A 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 28 • • 2 14 4 7 • • 2 2 7 2 28 = 7 • 2 • 2 28 = 2 • 2 • 7 The prime factorization of 28 is 2 • 2 • 7, or 22 • 7 .
Check It Out: Example 2B 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 36 12 12 2 3 6 4 2 2 2 3 2 3 1 1 36 = 3 • 2 • 2• 3 36 = 3 • 3 • 2 • 2 The prime factorization of 36 is 3 • 2 • 2 • 3, or 32 • 23.
