Greatest Common Factor Definition

The highest factor that is common for two or more given numbers is called Greatest Common Factor in Mathematics. Simply, when we divide two or more numbers, we will get some common factors. Among those factors, the highest number will be termed GCF.


Read about Common Factors here.

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For example,when we divide 18 and 21, we get the following factors.

18 = 2 x 3 x 3
24 = 2 x 2 x 2 x 3

We can see that “2” and “3” are common factors for the above numbers. But “3” is the highest common factor that will be called GCF of these numbers.

In Mathematics, GCF is also called HCF because both terms represent the highest common factor of the given numbers. You can find the greatest common factor of at least two numbers.

What is the difference between LCM and GCF?

Being similar terms, it is common for students to get confused between LCM and GCF. Here is a brief discussion about these terms to have an idea about their difference. LCM stands for Least Common Multiple. It is the smallest common multiple of the given numbers.

On the other hand, GCF is the highest common factor instead of a multiple. It comes out when we divide the given numbers to find their factors. If we say that GCF and LCM are concerned with two opposite dimensions in Mathematics, it will be right.


Read about LCM, Factors, and Multiples here.

How to find the GCF of two numbers?

Finding GCF is not a difficult task if you know the division process. Here are the two methods through which you can find the greatest common factor of as many words as you want (at least two).

Prime Factorization

It is a process in which you will divide the given numbers with their prime factors only. When you have done with the prime factorization, you will get a list of common factors that you can compare to find the highest common factor. Here is the step-by-step process to follow:

  • Divide both numbers separately with their prime factors
  • Write their factors in the up and down manner
  • Encircle the common factors
  • Compare the common factors to find the highest one
  • It will be termed as the GCF of those given numbers


Read about Prime Factors and Prime Factorization here.

Example 1:

What is the GCF of 18 and 21?


First of all, we have to find the prime factors of the given numbers. In our division, the factors of these numbers are:

18 = 2 x 3 x 3
21 = 3 x 7

We can see that “3” is the only common factor for these two numbers. So, it will be called the greatest common factor of 18 and 21.

Example 2:

What is the HCF of 60 and 90?


We have found the following prime factors of these two numbers.

60 = 2 x 2 x 3 x 5
90 = 2 x 3 x 3 x 5

From the above list of factors, we can see that “2”, “3”, and “5” are three common factors for 60 and 90. But “5” is the highest one, so it will be the HCF of these numbers.

Long Division

Another method to find the greatest common factor of the given numbers is long division. It is a bit harder method than prime factorization. Let us show you some points here that can make the process much easier for you.

  • Divide the larger number by the smaller one
  • It will give you a reminder
  • Make that remainder the divisor for the dividend of the previous step (the smaller number)
  • Perform division to get the remainder again
  • Keep repeating the above two steps until you get “0” as your reminder
  • The final divisor for which you have got a “0” remainder will be the GCF of those given numbers


Read about a Divisor, Dividend, Remainder, and Long Division here.

Formula to find GCF when LCM is given

As mentioned above, GCF and LCM are related to two opposite dimensions i.e. Multiples and Factors. But they have also a relationship through which the formula is originated. If you have been given any of these quantities, you can find the other.

Here is the formula with which you can do this:

LCM x GCF = First term x Second term

Let us explain it with a solved example given below.

Example 3:

Find GCF of 6 and 8 if their LCM is 24.


We only have to put the given values in the above formula.

24 x GCF = 6 x 8 

To find GCF, we have to divide 24 on both sides of the equation.

GCF = (6 * 8) / 24

By solving the right side of the above equation, we get the following GCF.

GCF = 2

Fun Facts about GCF

  • GCF of the given number divides those numbers completely without a remainder.
  • The GCF of two prime numbers is always “1”.
  • No other factor of two or more numbers can be greater than GCF.


What do you mean by GCF?

GCF stands for the greatest common factor of two or more numbers.

Are GCF and HCF the same?

Yes, GCF and HCF are two terms used to represent the same number in Mathematics.

What is the difference between GCF and LCM?

GCF is the greatest common factor while LCM is the least common multiple.

Can we find GCF through the division method?

Yes, we can find GCF through the long division technique.

Can we find the GCF of a single number?

No, GCF is the common factor that is only possible when we have at least two numbers with at least one common factor.