# Greatest Common Factor of Monomials

How to find the greatest common factor (GCF) of two or more monomials in maths? Grade 11 maths questions are presented along with Detailed Solutions and explanations are included.

## What is the greatest common factor (GCF) of monomials?

The greatest common factor of two or more monomials is the greatest monomial that divides evenly into all these monomials. It is found by first finding the prime factorization
Prime Factorization of each monomial.

### Example 1

Find greatest common factor of the monomials 12 x and 18 x 2.
Solution
1) Write the prime factorization of the monomial 12
x : 12 x = 2 � 2 � 3x
2) Write the prime factorization of the monomial 18
x 2 : 18 x 2 = 23 � 3 � xx
The greatest of the common factor (GCF) of 12
x and 18 x 2 is the product of all common factors in the prime factorization above:
GCF(12 x,18 x 2) = 2 � 3 � x = 6 x

### Example 2

Find greatest common factor of the monomials 30x 2y 3, 42x 3y 2 and 18 x 2y 2
Solution
1) Write the prime factorization of the monomial 30
x 2y 3 = 2 � 3 � 5 � x � xy � y � y
2) Write the prime factorization of the monomial 42
x 3y 2 = 2 � 3 � 7 � x � x � x � y � y
3) Write the prime factorization of the monomial 18
x 2 y 2 = 2 � 3 � 3 � x � xy � y
The greatest of the common factor (GCF) of 30
x 2y 3, 42x 3y 2 and 18 x 2y 2 is the product of all common factors in the prime factorization above:
GCF(30x 2y 3, 42x 3y 2 and 18 x 2y 2) = 2 � 3 � x � x � y � y = 6 x 2 y 2

Detailed Solutions and explanations are included.
1. Find the greatest common factor of the monomials 36 x2 and 42 x3.
2. Find the greatest common factor of 45 x3, 60 x2 and 75 x4.
3. What is the greatest common factor of 50 x2y3 , 75 x2y2 and 125 x4y3?
4. a) Find the prime factorization of the monomials of 35 x3 y 2 and 42 x2 y 3.
b) Simplify the rational expression ( 35 x3 y 2 ) / (42 x2 y 3)
5. Detailed Solutions and explanations are included.