Roots and Powers of Algebraic Expressions

How to Find the Greatest Common Factor

If the factors of a number are the different numbers that you can multiply together to get that original number, then the greatest common factor of two numbers is just the biggest one that both have in common. See some examples of what I'm talking about here!

Don't Confuse the GCF

The greatest common factor is usually used when simplifying fractions, and it's one of those topics that you learn pretty early on in your education but can easily forget or mistake for a different math idea, mainly the least common multiple. But before we can talk about the greatest common factor, often written as the GCF, we first have to know what a plain old regular factor is.

What Is a Factor?

Simply put, the factors of a number are the smaller numbers that make up that original one. Saying that slightly more mathematically sounds like this: The factors of a number are the different numbers that you can multiply together to get that original number. But a lot of math topics are best shown with examples, and this is probably one of them.

Let's start by looking at the factors of 6. The factors of 6 are going to be 2 and 3, because 2 x 3 = 6. It's also true that 1 and 6 are factors, then, because 1 x 6 is also equal to 6. That gives us our full list for the factors of 6 as 1, 2, 3, and 6

How about the factors of 60? Well, I know I can always do 1 times the number, so that works for this. Also, 60 is even, so I know 2 works, and 2 x 30 = 60. If we try dividing 60 by 3, we get 20, so that means I can add 3 and 20 to the list. If we continue on up, we find that 4 and 15, 5 and 12, 6 and 10 work - but the next one that works is 10 and 6, and this is basically just a repeat of 6 and 10. So, at this point we can stop, because the rest of the multiplication problems we do are just all repeats of numbers we've already got on our list. That makes the factors of 60 all the different numbers you see here: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Quite a lot of them.

The Greatest Common Factor

So, now we're ready to talk about what this lesson is really on: the greatest common factor, or GCF. In order to find a GCF, we need to be looking at two numbers, say 10 and 22. Then, we simply ask ourselves, of the factors these two numbers have, which one's the biggest that they have in common?

Well, factors of 10 are 1 and 10 and 2 and 5, while the factors of 22 are 1 and 22 and 2 and 11. That makes the greatest common factor of 10 and 22 2, because it's the biggest number I see on both the lists. That's it!

Examples

Let's try a few more examples just to make sure you've got it. Maybe this one: find the GCF of 27 and 45. We'll start by listing out the factors of each of these numbers individually, just like we learned earlier.

Looking at 27 first, 1 will always work, so we can start there. 27 is odd, so 2 is not going to work, but we can do 3 x 9. The next one that works is 9 x 3, so you've started repeating, and we can stop. That makes our list for the factors for 27 pretty short - just 1, 3, 9 and 27.

Next with 45 - after we count 1 and 45, we can again rule 2 out, but 3 and 15 is good, 4 doesn't work, but 5 x 9 does, and the next one is 9 x 5, so we've hit our repeating point. That makes our list of factors of 45 what you see here: 1, 3, 5, 9, 15, 45. So, answering the original question, 'What is the GCF of these two numbers?', is as easy as picking out the biggest number that is on both of these lists. Looks like 9 is our winner!

Last example: Find the greatest common factor between 4 and 16. We again begin by writing out all the factors of these two numbers. For 4, we get 1, 2 and 4, while for 16 we get 1, 2, 4, 8 and 16. So, the greatest common factor of 4 and 16 is the biggest number that's on both lists. That's 4.

Notice that 4 was one of the original numbers from the problem. That's totally okay! Some people get a little freaked out that this isn't allowed and decide to go with 2 instead because it's the next one down on the list. Don't do that! It's okay if the GCF is one of the original numbers from the problem.

Lesson Review

To review: the factors of a number are the different numbers that you can multiply together to get that original number. The greatest common factor of two numbers is the biggest factor that they both have in common and is often written as the GCF for short. It's okay to have the GCF of two numbers be one of the original numbers itself. And finally, as a side note, the place in math where you'll use this the most often is when you're simplifying fractions.