Let’s take a look at finding the greatest common factor. But before we find the greatest common factor, we probably should talk about what the greatest common factor is. To define greatest common factor, let’s start here with the word “factor.” Factors are numbers that are multiplied together to produce another number. For example, with the number 12, two times six equals 12. Two and six are both factors of 12. Three and four are also factors of 12.
It’s also possible that you’ve heard of something called a factor tree. And that looks something like this. We use a factor tree when we are interested in finding prime factors. We’ll also use it today when we’re looking at finding greatest common factor.
Okay, so now that we’ve talked about factor, we can talk about greatest and common and then put them all together. The word “greatest” here means just what you think it means, the largest, the biggest, the highest in numerical value. And the word “common” here means shared by all the numbers that you are comparing. So when we find the greatest common factor, we’ll be comparing two or more numbers. And we want something that is shared by all of them.
When we put the definition together, we could say the greatest common factor is the largest shared number that can be divided into each number you’re comparing. We should also note that many times we see the abbreviated form GCF. And that means greatest common factor. Here’s an example.
Find the greatest common factor of 12 and 30.
I’m gonna start by making a factor tree of the number 12 and the number 30. 12 is divisible by two and six. Two times three equals six. And now I have all the prime factors of 12. Moving on to 30, three times 10 equals 30. Two times five equals 10. And those are the prime factors.
Once we find the factors of 12 and 30, we can start to see common factors emerge. For example, three and three are both factors of 12 and 30. The number two is a factor of 12 and a factor of 30. But for us, the key is that we’re looking for the greatest common factor. We can find the greatest common factor by taking all the shared factors and multiplying them together.
Since each number is divisible by two and three, we multiply two times three together to get six. Six is then our greatest common factor. So the greatest common factor of 12 and 30 is six.
Our next example asks for the greatest common factor of nine and 14. Following the same procedure, let’s set up factor trees for nine and 14. Nine is divisible by three. Its prime factors are three and three. We’re finished there. And for 14, we have two and seven. Both of those are prime numbers, so that’s all we can do for our factor tree.
Take a look at the factors of nine and 14. So something is happening here that is different from our first example. I know what you’re thinking, no common factors? So how does this question work if there are no common factors? But there actually is one number that is still a factor for both nine and 14. You can multiply nine and 14 by one. You can also divide nine and 14 by one. So the greatest common factor here is the number one. If numbers don’t share any other common factors, we know that they always share one.
To encourage public transportation, Leo wants to gift some of his friends envelopes with bus tickets and subway tickets in them. If he has 32 bus tickets and 80 subway tickets to split out equally among the envelopes and he wants to make sure no tickets are left over, what is the greatest number of envelopes Leo can make?
For this problem, we’re gonna need to use — you guessed it — the greatest common factor. The greatest common factor of 32 and 80 will help us determine exactly how many envelopes Leo can make, because it will tell us how many groups we can divide each of these things into equally.
We’ll start this process the same way we always do, finding the factors of 32 and 80. 32 is divisible by two and 16. Two times eight equals 16. Two times four equals eight. Two times two equals four. With 80, I’m gonna start with eight and 10. Eight times 10 equals 80. Two times four equals eight. Two times two equals four. Two times five equals 10.
Here are all of the factors. And now let’s figure out which ones are the common factors. Both 32 and 80 have four twos as their factors. Two to the fourth power is the greatest common factor of 32 and 80. So we need to find out what is two to the fourth power. Two to the fourth power equals 16. This tells us that Leo can make 16 envelopes. Leo can then give out 16 envelopes. And in those 16 envelopes, each envelope will have two bus passes and five subway tickets. They will all be equal. There will be seven things in each envelope: two bus tickets, five subway tickets.
Let’s look at one other type of example. This example has three numbers that we are comparing and trying to find the greatest common factor of. We are gonna use the same process. So no matter if there’s three or four or five or six numbers, you’re trying to find the greatest common factor of, you’re going to use the same procedure that we’ve been using. We’re gonna use factor trees to find the greatest common factor.
And so here we go with our factor trees. 10 times 11 is 110. Two times five equals 10. And now for 40, four times 10, four equals two times two. 10 equals two times five. 12 times 10 equals 120. Two times six equals 12. Two times five equals 10. Finally, two times three equals six.
Each of these numbers have two and five in common, which means their greatest common factor is 10. We can also see this in this part of each factor tree. Two times five equals 10. 10 is the greatest common factor of these three numbers. And now it’s your turn to go out and try and find the greatest common factor of some numbers.