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Welcome to the greatest common divisor

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or greatest common factor video.

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So just to be clear, first of all, when someone asks you

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what's the greatest common divisor of twelve and eight?

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Or they ask you what's the greatest common factor of twelve and eight?

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That's a c right there for common.

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I don't know why it came out like that.

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They're asking you the same thing.

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I mean, really a divisor is just a number that can divide into something,

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and a factor-- well, I think, that's also a number that can divide into something.

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So a divisor and a factor are kind of the same thing.

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So with that out of the way, let's figure out,

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what is the greatest common divisor

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or the greatest common factor of twelve and eight?

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Well, what we do is, it's pretty straightforward.

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First we just figure out the factors of each of the numbers.

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So first let's write all of the factors out of the number twelve.

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Well, one is a factor. Two goes into twelve.

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Three goes into twelve.

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Four goes into twelve.

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Five does not to go into twelve.

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Six goes into twelve because two times six.

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And then, twelve goes into twelve of course.

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One times twelve.

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So that's the factors of twelve.

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Let's write the factors of eight.

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Well, one goes into eight.

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Two goes into eight.

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Three does not go into eight.

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Four does go into eight.

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And then the last factor, pairing up with the one is eight.

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So now we've written all the factors of twelve and eight.

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So let's figure out what the common factors of twelve and eight are.

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Well, they both have the common factor of one.

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And that's really not so special.

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Pretty much every whole number

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or every integer has the common factor of one.

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They both share the common factor two

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and they both share the common factor four.

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So we're not just interested in finding a common factor,

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we're interested in finding the greatest common factor.

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So all the common factors are one, two and four.

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And what's the greatest of them?

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Well, that's pretty easy.

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It's four.

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So the greatest common factor of twelve and eight is four.

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Let me write that down just for emphasis.

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Greatest common factor of twelve and eight equals four.

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And of course, we could have just as easily had said,

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the greatest common divisor of twelve and eight equals four.

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Sometimes it does things a little funny.

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Let's do another problem.

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What is the greatest common divisor of twenty-five and twenty?

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Well, let's do it the same way.

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The factors of twenty-five?

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Well, it's one.

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Two doesn't go into it.

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Three doesn't go into it.

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Four doesn't go into it.

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Five does.

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It's actually five times five.

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And then twenty-five.

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It's interesting that this only has three factors.

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I'll leave you to think about why this number only has three factors

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and other numbers tend to have an even number of factors.

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And then now we do the factors of twenty.

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Factors of twenty are one, two, four, five, ten, and twenty.

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And if we just look at this by inspection we see,

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well, they both share one, but that's nothing special.

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But they both have the common factor of?

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You got it-- five.

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So the greatest common divisor or greatest common factor

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of twenty-five and twenty, well, that equals five.

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Let's do another problem.

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What is the greatest common factor of five and twelve?

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Well, factors of five?

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Pretty easy.

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One and five.

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That's because it's a prime number.

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It has no factors other than one and itself.

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Then the factors of twelve?

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Twelve has a lot of factors.

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It's one, two, three, four, six, and twelve.

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So it really looks like only common factor they share is one.

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So that was, I guess, in some ways kind of disappointing.

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So the greatest common factor of five and twelve is one.

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And I'll throw out some terminology here for you.

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When two numbers have a greatest common factor of only one,

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they're called relatively prime.

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And that kind of makes sense because a prime number

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is something that only has one and itself as a factor.

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And two relatively prime numbers

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are numbers that only have one as their greatest common factor.

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Hope I didn't confuse you.

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Let's do another problem.

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Let's do the greatest common divisor of six and twelve.

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I know twelve's coming up a lot.

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I'll try to be more creative when I think of my numbers.

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Well, the greatest common divisor of six and twelve?

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Well, it's the factors of six.

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Are one, two, three, and six.

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Factors of twelve: one, two, three--

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we almost should have these memorized by now.

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Three, four, six, and twelve.

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Well, it turns out one is a common factor of both.

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Two is also a common factor of both.

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Three is a common factor of both.

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And six is a common factor of both.

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And of course, what's the greatest common factor?

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Well, it's six.

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And that's interesting.

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So in this situation the greatest common divisor--

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and I apologize that I keep switching between divisor and factor.

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The mathematics community should settle on one of the two.

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The greatest common divisor of six and twelve equals six.

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So it actually equals one of the numbers.

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And that makes a lot of sense

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because six actually is divisible into twelve.

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Well, that's it for now.

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Hopefully you're ready to do the greatest common divisor

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or factor problems.

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I think I might make another module in the near future

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that'll give you more example problems.