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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.

Procedure — To Find the Greatest Common Factor (GCF) of a Set of Numbers
Step 1 Write the prime factorization of each number.
Step 2 List each common prime factor the LEAST number of times it appears in any factorization.
Step 3 Multiply the prime factors in the list. If two numbers have no common prime factor, then their GCF is 1.

 

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.

 
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