Pre-Algebra: Finding Greatest Common Factors (GCF)

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FINDING THE GCF

So, we talked about factors. Suppose you have two numbers and look at both of those factors for those two numbers individually. Well, if they share a factor in common, then that’s called a common factor. And the biggest factor that the two numbers share together is called the greatest common factor.

Now, there are actually a lot of ways of computing, finding the greatest common factor if you are given a couple of numbers, even if you are given more than two numbers. I want to show you a few different ways right now.

The first way I want to show you, to illustrate it with the question, let’s find the greatest common factor of 8 and 12. That means I want to find a factor of 8 that is also a factor of 12 and also the biggest common factor of these numbers. So, one way is just to list all the factors of 8, all the factors of 12 and find the biggest common factor that they have in common. So, let’s try it. We list all the factors of 8. Well, 1, 2, 4 and 8. Those are all the factors of 8.

Let’s now list all the factors of 12. So, there is 1. So, we already see a common factor. So, this is good and if there are no others, then we know the greatest common factor will be 1. But notice, 2 is also a factor of 12. Now, we see that in fact, 2 is
even a bigger common factor than 1. So, let’s keep going. We see 3 is a factor of 12. 4 is a factor of 12. Six is a factor of 12 and 12 is a factor of 12.

Now, we are looking for the greatest factor that both of these have in common and we see that 1 is a common factor. 2 is a common factor. 3 is not, does not appear on this list. 4 is a common factor. 6 is not, does not appear here. 8 is not because it does not appear here and 12 is not because it does not appear here. So, 4 is the greatest common factor. Therefore, the greatest common factor, sometimes written as GCF, of these two numbers will be 4.

So, one way of doing this is to list all the factors and find the biggest one. That’s great for small numbers like 8 and 12 but when the numbers get really big, listing all the factors is kind of a pill.

There are others techniques, and I want to show you a couple of them.

So, let’s find the greatest common factor of these three numbers: 10, 15 and 30. Now, I am going to do it by looking at their prime factorizations.

So, if I now, first of all, factor these numbers into their prime factorization, 10 is going to be 2 times 5. 15 is going to be 3 times 5. Finally, 30 is going to be 2 times 3 times 5. I want to find the product of all the numbers that are in common with all
of these factorizations. So, what numbers appear in all 3 factorizations?

When I compare the 30 and 10, I see that 2 and 5 appear in both lists. That’s good but when I add on 15, there is no 2 but there is a 5. So, the only number that is a common factor of 10, 15 and 30 seems to be 5. So, you can sort of pluck off all the factors that are in common. In this case, it is just 5. So, I see the GCF is equal to 5. The greatest common factor of these three numbers is 5. That’s a neat way of doing it by writing the prime factorization. There is even another method that you can use sometimes.

Let’s try to find the greatest common factor for these three numbers: 18, 36 and 60. Now, I am going to use a ladder diagram idea to see if we can actually find these common factors and find the greatest one.

So, the way I do this is the following. I take the three numbers and write them down. Now, I look at them and see if I can see a number that is a common factor in each of them. And here is what I do. I know that 3 divides this number. 3 is the factor of this number. Three is the factor of this number and 3 is the factor of this number so 3 is the common factor.

So, I start setting up my ladder diagram with three numbers at the same time, and I write 3 out here. Now, I use the division to see what the quotient is. Here, I am going to get 6. Here, I am going to get 12 and here I am going to get 20. Now, look
at these new three numbers and ask, “Is there is a common factor that they all share?” Well, I see they are all even. Therefore, 2 is a common factor. So, I set this up again. Sort of fun! Little ladder, I guess, should I get a little triple ladder?
You have a triple bypass. That’s a medical thing. This is triple ladder, this is a math thing.

Now, we pull out the 2 and do the division. Here, I am left with a 3. Here, I am left with a 4 and here, I am left with a 10. Let’s see if this is all good here. So, I am pulling out a 2. Can you believe any of that? I actually made a little typo to see if
you are following all along.

Let’s see, you take the 2, divide it in here, and you see a 3. That’s good. What about this? You take 2. Take 12, divide by 2, is that really 4? No, I don’t think so. Why didn’t you say something? All right. I am going to let it pass this time. What should
it be? It should be a 6; we do the division. Take 12, divide it by 2, we get 6.

Now, what about the 20? Here, we do. In fact, you get 10. So, a little typo, it is ok to make mistakes. Never worry about making mistakes but you got to be able to find them. You’ve got to tell me next time. All right, good.

Now, let’s take a look at these three numbers: 3, 6 and 10. Do they share a common factor? Well, these two are even so, they share a factor of 2 but this one is not even, so that’s no good. These two share 3 but there is no factor of 3 in there. This has a 5, there is no 5. So, it looks like no. Of course, there is one, always 1, but in this case, it turns out that 1 is the greatest common factor of all three of these numbers and we can see that. No number bigger than 1 will divide in to each of these numbers at the same time. So, this tells me the game is over. These numbers actually have no common factor other than 1. So, I stop there. Then, I look along my ladder and 3 times 2, which is 6, gives me the greatest common factor. The greatest common factor will be the product of the numbers in the ladder here on the left. In this case, it is 6. Neat!

So, there is a great way to produce greatest common factors. There is this ladder method, there is the factorization method and there is also just the listing of the factors method. Find the greatest common factor. Pretty cool stuff!