Basic Math and Pre-Algebra

PART 1. The World of Numbers

CHAPTER 4. Factors and Multiples

Greatest Common Factor

The greatest common factor, or GCF, of two numbers is the largest number that is a factor of both. The greatest common factor of 21 and 24 is 3. The factors of 21, other than 1 and 21, are 3 and 7. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The largest number on both lists is 3.

DEFINITION

The greatest common factor (GCF) of two numbers is the largest number that is a factor of both.

To find the greatest common factor of 70 and 105, first find the factors of 70. The factors of 70 are 1, 2, 5, 7, 10, 14, 35 and 70. Then find the factors of 105. Those are 1, 3, 5, 7, 15, 21, 35, and 105. The largest number that shows up on both factor lists is 35, so 35 is the GCF of 70 and 105.

There are times when one of the numbers is the GCF of the two numbers. The GCF of 8 and 32 is 8, which is a factor of both 32 and of 8, because every number is a factor of itself. The factors of 8 are 1, 2, 4, and 8, and the factors of 32 are 1, 2, 4, 8, 16, and 32. 8 is the largest number that appears on both lists.

At the other extreme, sometimes the only factor the two numbers will have in common is 1. Remember 1 is a factor of every number. Numbers that have only 1 as a common factor are called relatively prime. The numbers 12 and 49 are relatively prime because the factors of 12 are 1, 2, 3, 4, 6, and 12, and the factors of 49 are 1, 7, and 49. The only number on both lists is 1.

DEFINITION

Two numbers are relatively prime if the only factor they have in common is 1.

For small numbers, making a list of factors isn’t difficult. If the number is small, the list of factors can’t be too long. But larger numbers can have long factor lists, so it’s important to have a more efficient method of finding the greatest common factor. That’s when you want to turn to the prime factorization.

To find the greatest common factor of 128 and 144, first work out the prime factorization of each number. Factor trees may help you factor to primes. The prime factorization of 128 is seven twos.

128 = 27 = 2 x 2 x 2 x 2 x 2 x 2 x 2

The prime factorization of 144 has both twos and threes.

144 = 24 x 32 = 2 x 2 x 2 x 2 x 3 x 3

Once you have both prime factorizations, look for any primes that appear in both (these numbers are underlined). It’s okay to have repeats if the factor repeats in both factorizations. In this case, there are four 2s that appear in both factorizations. The GCF of 128 and 144 is 2 x 2 x 2 x 2, which equals 24 or 16.

You can find the GCF of 210 and 495 by starting with the prime factorization of each number.

210 = 2 x 3 x 5 x 7

495 = 3 x 3 x 5 x 11

Both factorizations have a 3 and a 5. There’s only one 3 that’s common to both, so only one 3 goes into the prime factorization. 495 has a second 3, but 210 doesn’t share it. The GCF of 210 and 495 is 3 x 5, or 15.

CHECK POINT

Find the greatest common factor of each pair of numbers.

16. 18 and 42

17. 42 and 70

18. 144 and 242

19. 630 and 945

20. 286 and 715