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 \( 12x \) and \( 18x^2 \).

Solution

1) Write the prime factorization of the monomial \( 12 x \) : \( \quad 12 x = \color{red}{2} \cdot \color{black}{2} \cdot \color{red}{3} \cdot \color{red}{x} \)
2) Write the prime factorization of the monomial \( 18 x^2 \) : \( \quad 18 x^2 = \color{red}{2} \cdot \color{red}{3} \cdot \color{black}{3} \cdot \color{red}{x}\cdot \color{black}{x} \)
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: \[ \text{GCF} (12 x , 18 x^2) = \color{red}{2 \cdot 3 \cdot x = 6 x } \]

Example 2

Find greatest common factor of the monomials: \[ 30x^2 y^3,\quad 42x^3 y^2,\quad \text{and } 18x^2 y^2 \]

Solution


1) Write the prime factorization of the monomial \( 30x^2 y^3 \): \[ 30x^2 y^3 = \color{red}{2 \times 3} \times \color{black} 5 \times \color{red} {{x \times x} \times \color{red}{y \times y} \times \color{black}y} \] 2) Write the prime factorization of the monomial \( 42 x^3 y^2 \) : \[ 42 x^3 y^2 = \color{red}{2 \cdot 3} \cdot \color{black} 7 \cdot \color{red}{x \cdot x} \cdot \color{black} {x} \cdot \color{red}{y \cdot y} \] 3) Write the prime factorization of the monomial \( 18 x^2 y^2 \) : \[ 18 x^2 y^2 = \color{red} { 2 \cdot 3 \cdot {\color{black}3} \cdot x \cdot x \cdot y \cdot y } \] The greatest of the common factor of the three monomials is the product of all common factors in the prime factorization above: \[ \text{GCF}( 30x^2 y^3 , 42 x^3 y^2 , 18 x^2 y^2 ) = \color{red}{ 2 \cdot 3 \cdot x \cdot x \cdot y \cdot y = 6 x^2 y^2} \]


Answer the following questions


Detailed Solutions and explanations are included.
  1. Find the greatest common factor of the monomials \( 36 x^2 \; \text{and} \; 42 x^3 \).
  2. Find the greatest common factor of \( 45 x^3 \; , \; 60 x^2 \; \text{and} \; 75 x^4 \).
  3. What is the greatest common factor of \( 50 x^2 y^3 \; , \; 75 x^2 y^2 \; \text{and} \; 125 x^4 y^3 \)?
  4. a) Find the prime factorization of the monomials of \( 35 x^3 y^2 \; \text{and} \; 42 x^2 y^3 \).
    b) Simplify the rational expression \( \dfrac{35 x^3 y^2 }{42 x^2 y^3} \)
    5. Detailed Solutions and explanations are included.

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