# Solutions to Questions on Greatest Common Factor of Monomials

The detailed solutions and full explanations to the questions on Greatest Common Factor of Monomials are presented.

## Questions with Solutions

1. Find the greatest common factor of the monomials 36 x2 and 42 x3.
Solution
Write the prime factorization of the monomial 36 x2
36 x2 = 2 × 2 × 3 × 3 × x × x
Write the prime factorization of the monomial 42 x3
42 x3 = 2 × 3 × 7 × x × x × x
The greatest common factor of 36 x2 and 42 x3 is
2 × 3 × x × x = 6 x 2

2. Find the greatest common factor of 45 x3, 60 x2 and 75 x4.
Solution
Write the prime factorization of the monomial 45 x3
45 x3 = 3 × 3 × 5 × x × x × x
Write the prime factorization of the monomial 60 x2
60 x2 = 2 × 2 × 3 × 5 × x × x
Write the prime factorization of the monomial 75 x4
75 x4 = 3 × 5 × 5 × x × x × x × x
The greatest common factor of 45 x3, 60 x2 and 75 x4 is
3 × 5 × x × x = 15 x 2

3. What is the greatest common factor of 50 x2y3 , 75 x2y2 and 125 x4y3?
Solution
Write the prime factorization of the monomial 50 x2y3
50 x2y3 = 2 × 5 × 5 × x × x × y × y × y
Write the prime factorization of the monomial 75 x2y2
75 x2y2 = 3 × 5 × 5 × x × x × y × y Write the prime factorization of the monomial 125 x4y3
125 x4y3 = 5 × 5 × 5 × x × x × x × x × y × y × y
The greatest common factor of 50 x2y3 , 75 x2y2 and 125 x4y3 is
5 × 5 × x × x × y × y = 25 x 2 y 2

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)
Solution
a) Write the prime factorization of the monomial 35 x3 y 2
35 x3 y 2 = 5 × 7 × x × x × x × y × y
Write the prime factorization of the monomial 42 x2 y 3
42 x2 y 3 = 2 × 3 × 7 × x × x × y × y × y
We now use the prime factorization to simplify the rational expression
( 35 x3 y 2 ) / (42 x2 y 3) = (5 × 7 × x × x × x × y × y) / ( 2 × 3 × 7 × x × x × y × y × y) = 5x / 6y