$Rules and Definitions$

The Greatest Common Factor

The greatest common factor of two or more whole numbers is the largest whole number that divides evenly into each of the numbers. There are two ways to find the greatest common factor.

The first method is to list all of the factors of each number, then list the common factors and choose the largest one.

Example:

Find the GCF of 36 and 54.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.

The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18

Although the numbers in bold are all common factors of both 36 and 54, 18 is the greatest common factor.

The second method for finding the greatest common factor is to list the prime factors, then multiply the common prime factors.

Example:

Let's use the same numbers, 36 and 54 again to find their greatest common multiple.

The prime factorization of 36 is 2 x 2 x 3 x 3

The prime factorization of 54 is 2 x 3 x 3 x 3

Notice that the prime factorizations of 36 and 54 both have one 2 and two 3s in common. So, we simply multiply these common prime factors to find the greatest common factor. Like this...