Understanding how to find the Greatest Common Factor (GCF) is a critical mathematical skill, particularly when you need to simplify fractions or factor algebraic expressions.
This page features a lesson on two different methods for finding the GCF, followed by Grade 7 practice questions. Each question includes a detailed, step-by-step solution to help you check your work and master the process.
The greatest common factor of two or more numbers is the greatest whole number that divides evenly into all of these numbers.
Example: Find the greatest common factor of the numbers 12 and 18.
Look for the common factors of 12 and 18: They both share 1, 2, 3, and 6. The greatest of these common factors is 6.
Example: Find the greatest common factor of the numbers 60 and 90.
Look for the prime factors they share in common: one 2, one 3, and one 5.
Multiply these shared prime factors together: \( \mathbf{2 \times 3 \times 5} = 30 \).
The GCF of 60 and 90 is 30.
Tip: You can use our Greatest Common Factor Calculator to check your final answers!
Find the GCF for the following sets of numbers. Expand the solution blocks to check your prime factorization and final answer.
Find the greatest common factor of 36 and 42.
First, find the prime factorization of 36 and 42:
\( 36 = \mathbf{2} \times 2 \times \mathbf{3} \times 3 \)
\( 42 = \mathbf{2} \times \mathbf{3} \times 7 \)
Identify the prime factors they have in common. Both share one 2 and one 3.
Multiply the common factors: \( \mathbf{2 \times 3} = 6 \)
The GCF of 36 and 42 is 6.
Find the greatest common factor of 45, 60, and 75.
Find the prime factorization for all three numbers:
\( 45 = \mathbf{3} \times 3 \times \mathbf{5} \)
\( 60 = 2 \times 2 \times \mathbf{3} \times \mathbf{5} \)
\( 75 = \mathbf{3} \times 5 \times \mathbf{5} \)
Identify the prime factors shared by all three numbers. They all share one 3 and one 5.
Multiply the common factors: \( \mathbf{3 \times 5} = 15 \)
The GCF of 45, 60, and 75 is 15.
What is the greatest common factor of 360 and 252?
Find the prime factorization for both large numbers:
\( 360 = \mathbf{2 \times 2} \times 2 \times \mathbf{3 \times 3} \times 5 \)
\( 252 = \mathbf{2 \times 2} \times \mathbf{3 \times 3} \times 7 \)
Identify the common prime factors. They share two 2s and two 3s.
Multiply the common factors: \( \mathbf{2 \times 2 \times 3 \times 3} = 36 \)
The GCF of 360 and 252 is 36.
What is the greatest common factor of 324, 666, and 756?
Find the prime factorization for all three numbers:
\( 324 = \mathbf{2} \times 2 \times \mathbf{3 \times 3} \times 3 \times 3 \)
\( 666 = \mathbf{2} \times \mathbf{3 \times 3} \times 37 \)
\( 756 = \mathbf{2} \times 2 \times \mathbf{3 \times 3} \times 3 \times 7 \)
Identify the prime factors shared by all three numbers. They all share one 2 and two 3s.
Multiply the common factors: \( \mathbf{2 \times 3 \times 3} = 18 \)
The GCF of 324, 666, and 756 is 18.
Part a) First, find the prime factorization of 12 and 16:
\( 12 = \mathbf{2 \times 2} \times 3 \)
\( 16 = \mathbf{2 \times 2} \times 2 \times 2 \)
The GCF of 12 and 16 is \( \mathbf{2 \times 2} = 4 \).
Part b) We can use the relationship between the numbers to find the new GCF quickly. Notice that:
\( 1200 = 12 \times \mathbf{100} \)
\( 1600 = 16 \times \mathbf{100} \)
Because both numbers are scaled up by exactly 100, their Greatest Common Factor scales up by 100 as well.
Multiply the GCF from part (a) by 100: \( 4 \times 100 = 400 \).
The GCF of 1200 and 1600 is 400.