Main content

# Least common multiple of three numbers

CCSS.Math:

This least common multiple example gives us 3 numbers from which to find the lcm. This is a challenge, but fun. Do it with us! Created by Sal Khan and Monterey Institute for Technology and Education.

## Video transcript

What is the least common
multiple, abbreviated as LCM, of 15, 6, and 10? So the least common
multiple is exactly what the word is saying. It's the least common
multiple of these numbers. And I know that probably
didn't help you much. But let's actually work
through this problem. So to do that, let's just think
about the different multiples of the 15, 6, and 10 and
then find at the smallest multiple, the least multiple,
they have in common. So let's find the
multiples of 15. So you have 1 times 15 is 15. 2 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
we might have to go further. But I'll stop there for now. So that's the multiples
of 15 up through 105. Obviously, we can
keep going from there. Now, let's do the
multiples of 6. 1 times 6 is 6. 2 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 two
of them over here. We have a 30, and we have a 30. We have a 60 and a 60. So the smallest common
multiple, so if we only cared about the least
common multiple of 15 and 6, we would say it's 30. So let me write this
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's the smallest of all
of their common multiples. 60 is also a common multiple. But it's a bigger one. This is the least
common multiple. So this is 30. Well, we haven't thought
about the 10 yet. So let's bring the 10 in there. And I think you already
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's the smallest common
multiple of all of them. So it's 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 look at the
multiples of each of the number and then see what the
smallest multiple they have is in common. Another way to do
that is to look at the prime factorization
of each of these numbers. And the least common
multiple is the number that has all of the elements
of the prime factorizations of these and nothing else. So let me show you
what I mean by that. So you could do it this way. Or you could say 15 is the
same thing as 3 times 5. And that's it. That's its prime factorization. 15 is 3 times 5. Both 3 and 5 are prime. We can say that 6 is the
same thing as 2 times 3. That's it. That's its prime factorization. Both 2 and 3 are prime. And then we can say that 10 is
the same thing as 2 times 5. Both 2 and 5 are prime. So we're done factoring it. And so the least common
multiple of 15, 6, and 10 just needs to have all of
these prime factors. And what I mean, to be clear, is
in order to be divisible by 15, it has to have at
least one 3 and one 5 in its prime factorization. So it has to have at least
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's all we want. We just need one 3. So one 2 and one
3, this 2 times 3, ensures that we
are divisible by 6. And let me make it clear. This right here is the 15. And then to make sure that
we're divisible by 10, we have to have at
least one 2 and one 5. These two over here make sure
that we are divisible by 10. And so we have all of them. This 2 times 3 times 5 has
all of the prime factors of either 10, 6, or 15. So it is the least
common multiple. And so if you multiply this out,
you will get 2 times 3 is 6. 6 times 5 is 30. So either way, hopefully, both
of these resonate with you. And you see why they make sense. This second way is
a little bit better if you're trying to do it for
really complex numbers, numbers where you might have to be
multiplying it for a long time. Either way, both of
these are valid ways of finding the least
common multiple.