Definition of Greatest Common Factor
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Welcome to our Definition of Greatest Common Factor page.

As well as a clear definition of the greatest common factor we have links to our Greatest Common Factor calculator and also our Greatest Common Factor worksheets pages.

Definition of Greatest Common Factor

First of all let us think about what a factor is.

A factor is a number which divides into another number with no remainder.

In other words, any two numbers that multiply to give a product are both factors of the product.

  • 12 = 3 x 4, so 3 and 4 are both factors of 12.
  • 25 = 5 x 5 so 5 is a factor of 25
  • 60 = 12 x 5, so 5 and 12 are both factors of 60.

You will also note that 1 is a factor of all integers.

A common factor of two numbers is a factor that both numbers have in common.

The Definition of Greatest Common Factor of two numbers is the largest number that is a factor of both numbers.

And the Greatest Common Factor of a group of of numbers, is the largest factor that all the numbers have in common.

The Greatest Common Factor is the same as the Highest Common Factor, and the Greatest Common Divisor.

It is often abbreviated to the GCF.

If there are no other common factors, then the gcf is 1 (as 1 is a factor of all positive integers.)

If the greatest common factor of two integers is 1, the integers are said to be coprime.

Definition of Greatest Common Factor (or GCF) - How to find it!

So how can we find the greatest common factor?

There are a number of different ways to do this, but we are going to show you two of the easiest ones:

  • listing all the factors method
  • prime factorization method

Finding the Greatest Common Factor (or GCF) - by Listing the Factors Method

This is probably the simplest way to find the gcf of any set of numbers.

You just write out all the factors of each number in a list, and then find the largest number on all of the lists.

The only problem with this method is that it is sometimes easy to miss factors out.

Examples

Example 1) Find the greatest common factor of 24 and 32.

We list the factors of both numbers:

  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Factors of 32: 1, 2, 4, 8, 16, 32

The common factors of 24 and 32 are: 1, 2, 4, and 8.

The Greatest Common Factor is 8.


Example 2) Find the greatest common factor of 60 and 48.

We list the factors of both numbers:

  • Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  • Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The common factors of 60 and 48 are: 1, 2, 3, 4, 6, and 12.

The Greatest Common Factor is 12.


Example 3) Find the greatest common factor of 35 and 22.

We list the factors of both numbers:

  • Factors of 35: 1, 5, 7, 35
  • Factors of 22: 1, 2, 11, 22

The only common factor of 35 and 22 is 1.

The Greatest Common Factor is 1. This means that the two numbers are coprime.


Example 4) Find the greatest common factor of 39 and 13.

We list the factors of both numbers:

  • Factors of 39: 1, 3, 13, 39
  • Factors of 13: 1, 13 (this number is prime)

The common factors of 39 and 13 are 1 and 13.

The Greatest Common Factor is 13.

Note that it is fine for the greatest common factor to be one of the numbers you are testing, if the other number is a multiple of it like in the example shown.

Finding the Greatest Common Factor (or GCF) - by Prime Factorization Method

If you are using the prime factorization method, you first need to write out each number as a product of prime numbers.

If the number is prime to begin with, just write it down.

Now look at your lists of factors in each product and keep any that are in both lists.

If there is only one factor left then that is the gcf, otherwise multiply the factors together to get the gcf.

If there are no factors in both lists, then the gcf is 1.

Examples

Example 1) Find the greatest common factor of 30 and 45.

As a product of prime numbers:

  • 30 = 2 x 3 x 5
  • 45 = 3 x 3 x 5

We can see that the repeated part of the product in both lists is: 3 x 5.

The Greatest Common Factor is 3 x 5 = 15.


Example 2) Find the greatest common factor of 42 and 27.

As a product of prime numbers:

  • 42 = 2 x 3 x 7
  • 27 = 3 x 3 x 3

We can see that the repeated part of the product in both lists is: 3.

The Greatest Common Factor is 3.


Example 3) Find the greatest common factor of 88 and 45.

As a product of prime numbers:

  • 88 = 2 x 2 x 2 x 11
  • 45 = 3 x 3 x 5

We can see that there are no repeated factors in both lists.

The Greatest Common Factor is 1.


Example 4) Find the greatest common factor of 24, 18 and 42.

As a product of prime numbers:

  • 24 = 2 x 2 x 2 x 3
  • 18 = 2 x 3 x 3
  • 42 = 2 x 3 x 7

We can see that the repeated part of the product in both lists is: 2 x 3.

The Greatest Common Factor is 2 x 3 = 6.


Greatest Common Factor Calculator

Our Greatest Common Factor calculator will tell you the highest common factor between 2 or more numbers.

It will also list the factors of each of the numbers and tell you whether they are coprime or not.

Greatest Common Factor Worksheets

We have a range of greatest common factor worksheets with numbers up to 100.

Some of our sheets using the listing common factors method.

Some of our sheets use the prime factorization method.

We also have some trickier sheets where you can select the method you want.

Looking for some easier factor and multiples sheets?

We have worksheets to help you understand more about factors and multiples.

They are aimed at a 4th/5th grade level.

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Least Common Multiple Calculator

Our Least Common Multiple Calculator will find the lowest common multiple of 2 or more numbers.

It will also show you the working out using a choice of two different methods.

There are also some worked examples on the page with explanation of how it works.

Least Common Multiple Worksheets

We have a range of worksheets on how to find the least common multiple of two or three numbers.

The sheets vary in difficulty, and are suitable for 6th grade and up.

What is a Prime Number?

We have lots more information and help about prime numbers.

Prime numbers are very important in the properties they have, and you can find out more about them on this page.

Sieve of Erastosthenes

The Sieve of Erastosthenes is a method for finding what is a prime numbers between 2 and any given number.

Eratosthenes was a Greek mathematician (as well as being a poet, an astronomer and musician) who lived from about 276BC to 194BC.

If you want to find out more about his sieve for finding primes, and print out some Sieve of Eratosthenes worksheets, use the link below.

 

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