Greatest Common Factor (GCF)

Greatest Common Factor (GCF)

A factor is a number which divides another number without leaving a remainder. Every number is a factor of itself. The greatest common factor of two or more numbers is the largest common factor of the given numbers.

The Greatest Common Factor (GCF) of three whole numbers

The Greatest Common Factor of three numbers is the largest common factor among the common factors of three given numbers. There are several methods of finding the GCF of numbers. One of the methods is to use the factors of the given numbers. The other method is to use all prime factors of the given numbers.

Find the Greatest Common Factor of 6, 12 and 20.

Solution

By listing all factors of the given numbers

List all factors of 6, 12 and 20 as follows:

• Factors of 6 are 1, 2, 3 and 6.
• Factors of 12 are 1, 2, 3, 4, 6 and 12.
• Factors of 20 are 1, 2, 4, 5, 10 and 20.

Common factors are 1 and 2.

The Greatest Common Factor is 2.

Therefore, the GCF of 6, 12 and 20 is 2.

Alternative solution

By using prime factors of given numbers

Find all prime factors of 6, 12 and 20.

The prime factors of 6, 12 and 20 can be found by using tree diagrams as follows:

Thus, 2 is the greatest factor appearing in all the three numbers.

Therefore, the GCF of 6, 12 and 20 is 2.

Example

Find the GCF of 12, 18 and 15.

Solution

By listing all factors of the given numbers

List all factors of 12, 18 and 15.

• Factors of 12 are 1, 2, 3, 4, 6 and 12.
• Factors of 15 are 1, 3, 5 and 15.
• Factors of 18 are 1, 2, 3, 6, 9 and 18.

Common factors are 1 and 3.

The greatest common factor is 3.

Therefore, the GCF of 12, 18 and 15 is 3.

Alternative solution

By using the method of prime factors

It is easy to use tree diagrams to find the prime factors of 12, 18 and 15 as follows:

Thus, the prime factors of 12, 18 and 15 are:

• 12 = 2 × 2 × 3
• 15 = 3 × 5
• 18 = 2 × 3 × 3

Thus, 3 is the greatest factor appearing in all the three numbers.

Therefore, the GCF of 12, 18 and 15 is 3.