Greatest Common Divisor

4 example problems of determining the greatest common factor of two numbers by factoring the 2 numbers first

Welcome to the greatest common divisor

or greatest common factor video.

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

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

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

That's a c right there for common.

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

They're asking you the same thing.

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

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

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

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

what is the greatest common divisor

or the greatest common factor of twelve and eight?

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

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

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

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

Three goes into twelve.

Four goes into twelve.

Five does not to go into twelve.

Six goes into twelve because two times six.

And then, twelve goes into twelve of course.

One times twelve.

So that's the factors of twelve.

Let's write the factors of eight.

Well, one goes into eight.

Two goes into eight.

Three does not go into eight.

Four does go into eight.

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

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

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

Well, they both have the common factor of one.

And that's really not so special.

Pretty much every whole number

or every integer has the common factor of one.

They both share the common factor two

and they both share the common factor four.

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

we're interested in finding the greatest common factor.

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

And what's the greatest of them?

Well, that's pretty easy.

It's four.

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

Let me write that down just for emphasis.

Greatest common factor of twelve and eight equals four.

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

the greatest common divisor of twelve and eight equals four.

Sometimes it does things a little funny.

Let's do another problem.

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

Well, let's do it the same way.

The factors of twenty-five?

Well, it's one.

Two doesn't go into it.

Three doesn't go into it.

Four doesn't go into it.

Five does.

It's actually five times five.

And then twenty-five.

It's interesting that this only has three factors.

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

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

And then now we do the factors of twenty.

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

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

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

But they both have the common factor of?

You got it-- five.

So the greatest common divisor or greatest common factor

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

Let's do another problem.

What is the greatest common factor of five and twelve?

Well, factors of five?

Pretty easy.

One and five.

That's because it's a prime number.

It has no factors other than one and itself.

Then the factors of twelve?

Twelve has a lot of factors.

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

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

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

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

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

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

they're called relatively prime.

And that kind of makes sense because a prime number

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

And two relatively prime numbers

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

Hope I didn't confuse you.

Let's do another problem.

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

I know twelve's coming up a lot.

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

Well, the greatest common divisor of six and twelve?

Well, it's the factors of six.

Are one, two, three, and six.

Factors of twelve: one, two, three--

we almost should have these memorized by now.

Three, four, six, and twelve.

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

Two is also a common factor of both.

Three is a common factor of both.

And six is a common factor of both.

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

Well, it's six.

And that's interesting.

So in this situation the greatest common divisor--

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

The mathematics community should settle on one of the two.

The greatest common divisor of six and twelve equals six.

So it actually equals one of the numbers.

And that makes a lot of sense

because six actually is divisible into twelve.

Well, that's it for now.

Hopefully you're ready to do the greatest common divisor

or factor problems.

I think I might make another module in the near future

that'll give you more example problems.