Greatest Common Factor Calculator

What this calculator does

So, you have two or more whole numbers and you want the biggest number that divides all of them evenly. That is the greatest common factor, and this tool finds it. Enter your numbers and it returns their greatest common factor.

You type the numbers into the box separated by commas or spaces, so it works for two numbers or a whole list at once. The greatest common factor also goes by the highest common factor and the greatest common divisor; they all mean the same thing.

How to use it

1. Enter your whole numbers, separated by commas or spaces, like 12, 18 or 12 18.
2. Press Calculate.

What the greatest common factor is

A factor of a number is any whole number that divides it with nothing left over. The number 12 has factors 1, 2, 3, 4, 6, and 12. A common factor of several numbers is one that appears in all their factor lists. The greatest common factor is simply the largest of those shared factors. For 12 and 18, the common factors are 1, 2, 3, and 6, so the greatest common factor is 6: the biggest number that divides both 12 and 18 exactly.

How this tool finds it

The tool works in the most direct way, the same way you would by hand. For each number you enter, it works out the complete list of factors, every whole number that divides it evenly. Then it looks across all those lists for the factors they have in common, and picks out the largest one. That largest shared factor is the answer. Doing it by full factor lists is straightforward and easy to follow, which suits a tool meant to make the idea clear.

Where the GCF is useful

The greatest common factor earns its keep in a few everyday places. The big one is simplifying fractions: to reduce a fraction to lowest terms, you divide the top and bottom by their greatest common factor, and it drops straight to simplest form. It also helps when you are splitting things into equal groups, since the greatest common factor tells you the largest group size that works for several quantities at once. Anywhere you need the biggest even split shared by a set of numbers, this is the value you want.

A worked example

Enter 12 and 18. The tool lists the factors of 12 as 1, 2, 3, 4, 6, 12, and the factors of 18 as 1, 2, 3, 6, 9, 18. The numbers in both lists are 1, 2, 3, and 6, and the largest of these is 6. So the greatest common factor of 12 and 18 is 6.

When the answer is 1

Sometimes the greatest common factor comes out as 1. This happens when the numbers share no factor bigger than 1, and it has a name: such numbers are called relatively prime, or coprime. It does not mean the numbers themselves are prime, only that they have no common building blocks. For example, 8 and 9 are relatively prime, since 8 is 2 times 2 times 2 and 9 is 3 times 3, with nothing shared. A greatest common factor of 1 is a real and common answer, not a sign that something went wrong.

Questions people ask

What is the greatest common factor?

The largest whole number that divides all your numbers evenly. For 12 and 18 it is 6.

Is it the same as the GCD or HCF?

Yes. Greatest common factor, highest common factor, and greatest common divisor are three names for the same thing.

How does the tool work it out?

It lists every factor of each number, finds the factors common to all of them, and picks the largest.

Why is it useful for fractions?

Dividing the top and bottom of a fraction by their greatest common factor reduces it to its simplest form in one step.

What if the answer is 1?

Then the numbers share no factor above 1 and are called relatively prime. It is a normal result.

References

On the greatest common divisor. The greatest common factor is the largest divisor shared by a set of integers, classically found by Euclid's algorithm.

1. Eric W. Weisstein, "Greatest Common Divisor," from MathWorld, a Wolfram resource, on the largest divisor common to two or more integers.
2. Euclid, Elements, Book VII, Proposition 2, D. E. Joyce's online edition, Clark University, on finding the greatest common measure of two numbers.