Greatest Common Divisor

Welcome to the greatest
common divisor or greatest

common factor video.

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

whether what's the greatest
common divisor of 12 and 8?

Or they ask you what's
the greatest common

factor of 12 and 8?

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 12 and 8?

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 12.

Well, 1 is a factor,
2 goes into 12.

3 goes into 12.

4 goes into 12.

5 does not to go into 12.

6 goes into 12
because 2 times 6.

And then, 12 goes
into 12 of course.

1 times 12.

So that's the factors of 12.

Let's write the factors of 8.

Well, 1 goes into 8.

2 goes into 8.

3 does not go into 8.

4 does go into 8.

And then the last factor,
pairing up with the 1 is 8.

So now we've written all
the factors of 12 and 8.

So let's figure out what the
common factors of 12 and 8 are.

Well, they both have the
common factor of 1.

And that's really
not so special.

Pretty much every whole
number or every integer has

the common factor of 1.

They both share the common
factor 2 and they both

share the common factor 4.

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 1, 2 and 4.

And what's the
greatest of them?

Well, that's pretty easy.

It's 4.

So the greatest common
factor of 12 and 8 is 4.

Let me write that down
just for emphasis.

Greatest common factor
of 12 and 8 equals 4.

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

greatest common divisor
of 12 and 8 equals 4.

Sometimes it does
things a little funny.

Let's do another problem.

What is the greatest common
divisor of 25 and 20?

Well, let's do it the same way.

The factors of 25?

Well, it's 1.

2 doesn't go into it.

3 doesn't go into it.

4 doesn't go into it.

5 does.

It's actually 5 times 5.

And then 25.

It's interesting that
this only has 3 factors.

I'll leave you to think about
why this number only has 3

factors and other numbers
tend to have an even

number of factors.

And then now we do
the factors of 20.

Factors of 20 are 1,
2, 4, 5, 10, and 20.

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

both share 1, but that's
nothing special.

But they both have the
common factor of?

You got it-- 5.

So the greatest common divisor
or greatest common factor of 25

and 20- well, that equals 5.

Let's do another problem.

What is the greatest common
factor of 5 and 12?

Well, factors of 5?

Pretty easy.

1 and 5.

That's because it's
a prime number.

It has no factors other
than 1 and itself.

Then the factors of 12?

12 has a lot of factors.

It's 1, 2, 3, 4, 6, and 12.

So it really looks like only
common factor they share is 1.

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

So the greatest common
factor of 5 and 12 is 1.

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

When two numbers have a
greatest common factor of

only 1, they're called
relatively prime.

And that kind of makes sense
because a prime number is

something that only has 1
and itself as a factor.

And two relatively prime
numbers are numbers that

only have 1 as their
greatest common factor.

Hope I didn't confuse you.

Let's do another problem.

Let's do the greatest common
divisor of 6 and 12.

I know 12's coming up a lot.

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

Well, the greatest common
divisor of 6 and 12?

Well, it's the factors of 6.

Are 1, 2, 3, and 6.

Factors of 12: 1, 2, 3--
we should have these

memorized by now.

3, 4, 6, and 12.

Well, it turns out 1 is a
common factor of both.

2 is also a common
factor of both.

3 is a common factor of both.

And 6 is a common
factor of both.

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

Well, it's 6.

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 6 and 12 equals 6.

So it actually equals
one of the numbers.

And that makes a lot of
sense because 6 actually

is divisible into 12.

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.