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 × 3 × x
2) Write the prime factorization of the monomial 18 x^{ 2} : 18 x^{ 2} = 2 × 3 × 3 × x × x
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^{ 2}y^{ 3}, 42x^{ 3}y^{ 2} and 18 x^{ 2}y^{ 2}
Solution
1) Write the prime factorization of the monomial 30x^{ 2}y^{ 3} = 2 × 3 × 5 × x × x × y × y × y
2) Write the prime factorization of the monomial 42x^{ 3}y^{ 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 × x × y × y
The greatest of the common factor (GCF) of 30x^{ 2}y^{ 3}, 42x^{ 3}y^{ 2} and 18 x^{ 2}y^{ 2} is the product of all common factors in the prime factorization above:
GCF(30x^{ 2}y^{ 3}, 42x^{ 3}y^{ 2} and 18 x^{ 2}y^{ 2}) = 2 × 3 × x × x × y × y = 6 x^{ 2} y^{ 2}
Answer the following questions
Detailed Solutions and explanations are included.

Find the greatest common factor of the monomials 36 x^{2} and 42 x^{3}.

Find the greatest common factor of 45 x^{3}, 60 x^{2} and 75 x^{4}.

What is the greatest common factor of 50 x^{2}y^{3} , 75 x^{2}y^{2} and 125 x^{4}y^{3}?

a) Find the prime factorization of the monomials of 35 x^{3} y^{ 2} and 42 x^{2} y^{ 3}.
b) Simplify the rational expression ( 35 x^{3} y^{ 2} ) / (42 x^{2} y^{ 3})
Detailed Solutions and explanations are included.