Least Common Multiple

Example of figuring out the least common multiple of two nunmbers

Hello, I'm now going to do some practice least common

multiple problems for you.

After I do a couple of these problems you should be able

to go to the least common multiple module and do

some of them yourselves.

Let's say the least common multiple of 10 and 8.

I'm going to show you two ways to do a least

common multiple problem.

One I call just the brute force method and I think it's good

because it'll give you a good sense of what least common

multiple is and then I'll also show you what I call the

more elegant method.

So the brute force method is literally just to write down

all the multiples of the two numbers and figure out

what the least common multiple they have is.

So let's write all the multiples of 10.

So 10 times 1 is 10.

10 times 2 is 20.

30, 40, 50, 60, whoops.

Not 67.

70, 80, 90, 100 and so on.

Multiples of 8 are 8, 16, 24, 32, 40, 48,

64, 72, 80 and so on.

So let's see.

Let's see if we can identify what the common multiples are.

Well, immediately I see that 10 times 4 is 40 and 8 times 5 is

also 40, so that's a common multiple.

If we keep going we see that 10 times 8 is 80 and 8

times 10 is also 80.

And if we were to keep going we would also see that

120 is a common multiple.

We'd see that 160 is a common multiple.

But out of the ones we listed 40 and 80 are

our common multiples.

And if we were to ask, what is the least common multiple?

Well, 40 is lower than 80, so we say 40 is the

least common multiple.

That's what I call the brute force method.

Now what I would say the elegant method is, is what you

do is you look at the factors of 10 and you say, well, the

factors of 10 are 1, 2, 5, and 10.

And the factors of 8 are 1, 2, 4, and 8.

And you say, what's the greatest common factor

of the two numbers?

Well, they all share the common factor one.

Every integer shares that common factor.

But the number 2.

They both share that common factor.

So what we can say is, is that the least common multiple of 10

and 8-- and this is the elegant way and it might not be obvious

to you why it works and I might do another module with you

to show you why this works.

But the least common multiple of two numbers is always equal

to the two numbers-- 8 times 10-- and the dot is this

another fancy way of writing times.

8 times 10 and then you divide that by the greatest

common factor of 8 and 10.

Well, 8 times 10 is 80, and the greatest common

factor of 8 and 10?

Well, we just figured that out.

That's 2.

So that equals 40.

In general, in my head, and you'll learn to do these

problems, in your head.

I tend to do it this first way.

I don't figure out what the greatest common factor is

and then I multiply the numbers and divide them.

Because for smaller numbers like 8 or 10 or two and 3, it's

pretty easy to just think about the multiples and figure out

the least common multiple.

But if you had really large numbers or if you're writing a

computer program, that had to deal with arbitrary numbers,

then you'd probably want to use the second method.

And if you're ever in doubt the second method always works just

to make sure you haven't overlooked some numbers in

using the method on the left.