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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 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.