# 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 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 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 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 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