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.

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

Find the greatest common factor of 45 x^{3}, 60 x^{2} and 75 x^{4}.
Solution
Write the prime factorization of the monomial 45 x^{3}
45 x^{3} = 3 × 3 × 5 × x × x × x
Write the prime factorization of the monomial 60 x^{2}
60 x^{2} = 2 × 2 × 3 × 5 × x × x
Write the prime factorization of the monomial 75 x^{4}
75 x^{4} = 3 × 5 × 5 × x × x × x × x
The greatest common factor of 45 x^{3}, 60 x^{2} and 75 x^{4} is
3 × 5 × x × x = 15 x^{ 2}

What is the greatest common factor of 50 x^{2}y^{3} , 75 x^{2}y^{2} and 125 x^{4}y^{3}?
Solution
Write the prime factorization of the monomial 50 x^{2}y^{3}
50 x^{2}y^{3} = 2 × 5 × 5 × x × x × y × y × y
Write the prime factorization of the monomial 75 x^{2}y^{2}
75 x^{2}y^{2} = 3 × 5 × 5 × x × x × y × y
Write the prime factorization of the monomial 125 x^{4}y^{3}
125 x^{4}y^{3} = 5 × 5 × 5 × x × x × x × x × y × y × y
The greatest common factor of 50 x^{2}y^{3} , 75 x^{2}y^{2} and 125 x^{4}y^{3} is
5 × 5 × x × x × y × y = 25 x^{ 2} y^{ 2}

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

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