Least Common Multiple (LCM)

U02_L3_T1_we6 Least Common Multiple (LCM)

What is the least common multiple, abbreviated as LCM, of 15, 6 and 10?

So the LCM is exactly what the word is saying, it is the least common multiple of these numbers.

And I know that probably did not help you much. But lets actually work trough this problem.

So to do that, lets think of the different multiples of 15, 6 and 10.

and then find the smallest multiple, the least multiple they have in common.

So, lets find the multiples of 15. You have: 1 times 15 is 15, two times 15 is 30,

then if you add 15 again you get 45, you add 15 again you get 60, you add 15 again,

you get 75, you add 15 again, you get 90, you add 15 again you get 105.

and if still none of these are common multiples with these guys over here

then you may have to go further, but I will stop here for now.

Now that's the multiples of 15 up through 105. Obviously, we keep going from there. Now lets do the multiples of 6.

Let's do the multiples of 6: 1 times 6 is 6, two times 6 is 12, 3 times 6 is 18, 4 times 6 is 24,

5 times 6 is 30, 6 times 6 is 36, 7 times 6 is 42, 8 times 6 is 48,

9 times 6 is 54, 10 times 6 is 60. 60 already looks interesting, because it is a common multiple of both 15 and 60. Although we have to of them over here.

We have 30 and we have a 30, we have a 60 and a 60. So the smallest LCM...

...so if we only cared about the least common multiple of 15 and 6.

We would say it is 30. Lets write that down as an intermediate: the LCM of 15 and 6. So the least common multiple,

the smallest multiple that they have in common we see over here. 15 times 2 is 30 and 6 times 5 is 30.

So this is definitely a common multiple and it is the smallest of all of their LCMs.

60 is also a common multiple, but it is a bigger one. This is the least common multiple. So this is 30.

We have not thought of the 10 yet. So lets bring the 10 in there. I think you see where this is going.

Let's do the multiples of 10. They are 10, 20, 30, 40... well, we already went far enough. Because we already got to 30,

and 30 is a common multiple of 15 and 6 and it is the smallest common multiple of all of them.

So it is actually the fact that the LCM of 15, 6 and 10 is equal to 30.

Now, this is one way to find the least common multiple. Literally, just find and look at the multiples of each of the numbers...

and then see what the smallest multiple is they have in common.

Another way to do that, is to look at the prime factorization of each of these numbers

and the LCM is the number that has all the elements of the prime factorization of these and nothing else.

So let me show you what I mean by that. So, you can do it this way or you can say that 15 is the

same thing as 3 times 5 and that's it. That is its prime factorization, 15 is 3 times 5, since both 3 and 5 are prime numbers.

We can say that 6 is the same thing as 2 times 3. That's it, that is its prime factorization, since both 2 and 3 are prime.

And then we can say that 10 is the same thing as 2 times 5. Both two and 5 are prime, so we are done factoring it.

So the LCM of 15, 6 and 10, just needs to have all of these prime factors.

And what I mean is... to be clear, in order to be divisible by 15

it has to have at least one 3 and at least one 5 in its prime factorization, so it needs to have one 3 and at least one 5.

By having a 3 times 5 in its prime factorization that ensures that this number is divisible by 15.

To be divisible by 6 it has to have at least one 2 and one 3. So it has to have at least one 2 and we already have a 3 over here so that is all we want.

We just need one 3. So one 2 and one 3. That is 2 times 3 and ensures we are divisible by 6. And let me make it clear, this right here is the 15.

And then to make sure we are divisible by 10, we need to have at least one 2 and one 5. These two over here make sure we are divisible by 10.

and so we have all of them, this 2 x 3 x 5 has all of the prime factors of either 10, 6 or 15, so it is the LCM.

So if we multiply this out, you will get, 2 x 3 is 6, 6 x 5 is 30.

So either way. Hopefully these kind of resonate with you and you see why they make sense.

This second way is a little bit better, if you are trying to do it for really complex numbers...

...numbers, where you might have to be multiplying for a really long time.

Well, either way, both of these are valid ways of finding the least common multiple.