7.4 Greatest Common Factor- GCF (P.170-173)

I. There are two methods for finding the greatest common factor:

• List all factors and find the largest
• multiply the common prime factors

A.List and find largest
1.First, list all of the factors of each number
2.Then, list the common factors and choose the largest one.

B.List the prime factors, then multiply the common prime factors.
1. Find the prime factors for the numbers in question
2.Recognize the common prime factors
3. Multiply them together to find the greatest common factor

Example for list all factors:
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.

Example for multiplying common prime factors:
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...

2 x 3 x 3 = 18

Both methods for finding the greatest common factor work!

-examples from helpwithfractions.com