How to Get the Greatest Common Factor of Numbers

By | July 4, 2014

The numbers that can divide an integer is called its factor or divisor. For example, the factors of 4 are 1, 2, and 4 because these are the numbers that divide 4 without having a remainder. Another example is 6 which has factors 1, 2, 3, and 6. It is clear that each number has always 1 and itself as factors. Note that in this discussion, when I say number, I mean positive integer.

If we select more than one number, we can observe that they have common factors (just like having common multiples). Let’s have the following examples. 

How to Get the Greatest Common Factor of Numbers

Example 1: What are the common factors of 12 and 18?

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

If we examine the factors of 12 and 18, we see that there are 4 common factors: 1, 2, 3 and 6. Among the factors, 6 is the largest. Therefore, we say that 6 is the greatest common factor (GCF) or greatest common divisor (GCD) of 12 and 18. Example 2 : Find the GCF of 20, 32, 28.

Factors of 20: 1, 2, 3, 4, 5, 10, 20

Factors of 32: 1, 2, 4, 8, 16, 32

Factors of 28: 1, 2, 4, 7, 14, 28

As we can see, the common factors of 20, 32, and 28 are 1, 2, and 4. The GCD or GCF of the three numbers is 4.

Another way to get the greatest common factor of numbers is to write their prime factorization. Prime factorization is the process of expressing a number as product of prime numbers. A prime number is a number which is only divisible by 1 and itself (read Introduction to Prime Numbers if you don’t know what is a prime number). The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

We will use the examples above and use prime factorization in order to get their greatest common factor.

Example 3: Find the GCF of 12 and 18 using prime factorization.

Prime factorization of 12: 2 × 2 × 3

Prime Factorization of 18: 2 × 3 × 3

Now to get the greatest common factor, we multiply the common factors to both numbers. The common factors to both are 2 and 3, therefore, the greatest common factor of 12 and 18 is 2 × 3 = 6.

Example 4: Find the GCF of 12 and 18 using prime factorization.

Prime factorization of 20: 2 × 2 × 5

Prime factorization of 32: × 2 × 2 × 2

Prime factorization of 28: 2 × 2 × 2

In this example, 2 and 2 are common to all the three numbers, so the GCD or GCD of these three numbers is 2 × 2 which is equal to 4. 

The difference between the two methods is that in the first method, you list all the factors and find the largest number. In the second method, you list the prime factorization and the multiply the factors that are common to all numbers.

What’s the use of greatest common factor?

Well, GCF are used a lot in mathematics, but in the Civil Service Exam, you will use it when you reduce fractions to lowest terms. For example, your final answer is

\frac{12}{18}

and \frac{12}{18} is not on the choices. Then, you know that you have to get the greatest common factor of 12 and 18 and divide both the numerator and denominator by it. So, the answer is

\displaystyle \frac{12 \div 6}{18 \div 6} = \displaystyle \frac{2}{3}

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