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Greatest Common Factor
The greatest common factor. This is sometimes called the greatest common factor and sometimes the greatest common divisor. Both of those mean the same thing. I will use the abbreviation GCF throughout this video for greatest common factor. Finding the greatest common factor of two integers is an important skill in many contexts.
So first of all, we have to find out how to do it, and then we’ll talk about how to apply it in coming videos. So first of all, what is the greatest common factor of 24 and 40? What do we mean by asking the greatest common factor? Well, think about it this way. We could list the factors.
So the factors of 24, those are all the factors of 24. Those are all the factors of 40. Notice that some of those factors they have in common. They appear on both lists. Those are the common factors. So we can make a list of the common factors.
The common factors of 24 and 40 are 1, 2, 4, and 8. And on that list the greatest common factor, is just the highest number on that list. The largest number in the common factor list is 8. And therefore, 8 is the greatest common factor of 24 and 40. We could always list the factors, identify the common factors, and find the largest.
That is the greatest common factor. The problem is, as we have seen, often we have to deal with numbers that are larger. And it would be very time consuming simply to list the, all the factors. For example, if we were given two numbers up in the hundreds, it would take a very long time to list all the factors. Instead, we are going to use a procedure involving the prime factorizations.
This is one of the many handy uses of the prime factorizations. Suppose we want the greatest common factors of 360, and 800. Well, listing these factors would take forever. We don’t wanna go down that road. Instead, we’re gonna find the prime factorizations of both of those numbers. So, 360, obviously, that’s 36 times 10 or 6 times 6 times 10, and we can simplify that to 2 to the 3rd times 3 squared times 5.
800 is obviously 8 times 100 or 8 times 10 times 10. That simplifies to 2 to the 5th time 5 squared. All right, so now we have prime factorizations. Now here are the questions we’re gonna ask. What is the highest power of 2 they have in common? So, the first one has 3 factors of 2.
The second one has 5 factors of 2. And so that means that both of them have at least 3 factors of 2. So 3 factors of 2 is something that they both have. Each has at least 3 factors of 2. Similarly, what’s the highest power of 3 they have in common? Well, one of them has 2 powers of 3, but the other one has no powers of 3.
So 3 cannot appear in the common factor because one of the two numbers doesn’t have it in common. So that just has to be zero. The highest power of 5 in common, 1 has one power of 5, 1 has two powers of 5. So they each have at least one power of 5. Well, now we put that together.
The common factor has to have 3 factors of 2, 0 factors of 3, and 1 factor of 5. And when we multiply that out, 2 to the 5th, 2 to the 3rd is 8. And then 8 times 5 is 40, and that’s the greatest common factor. What is the greatest common factor of 720 and 1200? Pause the video here, and try this on your own. Then we’ll talk about the answer.
Okay. So, first of all, we find the greatest common factors. We find the prime factorizations of these. Those are the prime factorizations. Now we start looking at the individual factors. So, highest power of 2 in common?
They both have 2 to the 4th. That’s easy. Highest power of 3 in common? We have 3 to the 1st, 3 to the 2nd, so the highest power they have in common is 3 to the 1st. Highest power of 5 in common, well we have 5 to the 1st, 5 to the 2nd, so the highest power in common is 5 to the 1st.
Therefore, the common factor has to be 2 to the 4th times 3 to the 1st times 5 to the 1st. So in other words, 2 to the 4th times 3 times 5. And if we do a little rearranging, I’m actually gonna take one of those powers of 2 and multiply it by the 5. So I’m gonna rearrange like this, I’m going to leave 3 of the powers of 2 over on the left.
But put one of them next to the 5, and that allows me to do 2 times 5, which is 10. Of course, it’s easy to multiply by 10. 2 to the 3rd is 8. 3 times 8 is 24. So it’s 24 times 10, or 240, and that is the greatest common factor.
In this video, we talked about what the greatest common factor is. We find the common factors of the number. And of all the common factors the greatest common factor is the largest number on that common factor list. We talked about how to use the prime factorization of two numbers to find the greatest common factor.
And indeed, most of the times when you have to find the greatest common factor on the test, it will be larger numbers, and you will have to use the prime factorization method. Now, we haven’t talked at all about why you would want to find the greatest common factor. I’ll just point out here, we will talk about the single most important use of the greatest common factor in the next video.
We use the greatest common factor to find the least common multiple. So that’s coming up in the next video lesson.
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