Greatest Common Factor

Remember that a factor is a number that divides evenly into a given number.  A number that is a factor of two or more numbers is a common factor of those numbers. The greatest common factor (GCF) is the largest factor that those numbers have in common.

Listing the factors of each number is one way to find the GCF.  Another way to find the GCF uses the prime factorization of each number.

Example:  Find the GCF of 24 and 30.

We can list out the factors of each number and find the largest one they have in common.

Factors of 24:  {1, 2, 3, 4, 6, 8, 12, 24}

Factors of 30:  {1, 2, 3, 5, 6, 10, 15, 30}

We see their common factors are 1, 2, and 6.  The GCF is the largest one, which is 6.

To find the GCF of larger numbers, you can write out the prime factorization of each number. Next, identify the prime factors that each number has in common. The GCF is the product of these common factors.

Using our example once again:

24:  2 × 12

       2 × 2 × 6

       2 × 2 × 2 × 3

30:  2 × 15

       2 × 3 × 5

So the prime factorizations are

24:  2 × 2 × 2 × 3

30:  2 × 3 × 5

We see that both factorizations have a 2 and a 3, so the GCF of 24 and 30 is 2 × 3 = 6.

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