Finding the GCF of a Set of Monomials

Finding the GCF of a Set of Numbers

Recall that the greatest common factor (GCF) of a set of numbers is the greatest number that is a factor of all the numbers in the set.

Example

Find the GCF of -36, 72, and -90.

Solution

Step 1 Write the prime factorization of each number. Prime factorization applies to natural numbers, so first write each negative number as -1 times its opposite.	-36 -90	= -1 · 36 = -1 · 90
A factor tree may be helpful in finding the prime factorizations.
	-36 72 -90	= -1 =  2 = -1	· 2 · 2 · 3 · 3 · 2 · 2 · 3 · 3 · 2 · 3 · 3 · 5
Step 2 List each common prime factor the LEAST number of times it appears in any factorization.
The common prime factors are 2 and 3. The least number of times that 2 appears in a factorization is once. So, 2 appears once in the list. The least number of times that 3 appears in a factorization is twice. So, 3 appears twice in the list. Here is the list: 2, 3, 3
Step 3 Multiply the prime factors in the list. Thus, the GCF of -36, 72, and -90 is 18. To see that 18 is a common factor of -36, 72, and -90, we write each as a product using 18 as one of the factors.	2 · 3 · 3   -36 72	= 18   = 18 · (-2) = 18 · 4

We can use a similar procedure to find the GCF of a set of monomials that contain variables.

Procedure — To Find the Greatest Common Factor (GCF) of a Set of Monomials

Step 1 Write the factorization of each monomial.

Step 2 List each common factor the LEAST number of times it appears in any factorization.

Step 3 Multiply the factors in the list. If two monomials have no common factors, other than 1, then their GCF is 1.