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 â€”
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 Â· 36 = 1
Â· 90 
A factor tree may be helpful in finding the prime factorizations. 

36
72
90 
= 1 = 2
= 1 
Â· 2 Â· 2 Â· 3 Â· 3
Â· 2 Â· 2 Â· 3 Â· 3
Â· 2 Â· 3 Â· 3 Â· 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 Â· 3 Â· 3
36
72 
= 18
= 18 Â· (2)
= 18 Â· 4 
We can use a similar procedure to find the GCF of a set of monomials that
contain variables.
Procedure â€”
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.
