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# Least common multiple

Video transcript

What is the least common
multiple of 36 and 12? So another way to say this is
LCM, in parentheses, 36 to 12. And this is literally
saying what's the least common
multiple of 36 and 12? Well, this one might
pop out at you, because 36 itself
is a multiple of 12. And 36 is also a multiple of 36. It's 1 times 36. So the smallest number that is
both a multiple of 36 and 12-- because 36 is a multiple
of 12-- is actually 36. There we go. Let's do a couple more of these. That one was too easy. What is the least common
multiple of 18 and 12? And they just state this
with a different notation. The least common
multiple of 18 and 12 is equal to question mark. So let's think about
this a little bit. So there's a couple of ways
you can think about-- so let's just write down our
numbers that we care about. We care about 18,
and we care about 12. So there's two ways that
we could approach this. One is the prime
factorization approach. We can take the prime
factorization of both of these numbers
and then construct the smallest number
whose prime factorization has all of the ingredients
of both of these numbers, and that will be the
least common multiple. So let's do that. 18 is 2 times 9, which is
the same thing as 2 times 3 times 3, or 18 is 2 times 9. 9 is 3 times 3. So we could write 18 is
equal to 2 times 3 times 3. That's its prime factorization. 12 is 2 times 6. 6 is 2 times 3. So 12 is equal to
2 times 2 times 3. Now, the least common
multiple of 18 and 12-- let me write this down-- so
the least common multiple of 18 and 12 is going to have to have
enough prime factors to cover both of these
numbers and no more, because we want the least
common multiple or the smallest common multiple. So let's think about it. Well, it needs to have at
least 1, 2, a 3 and a 3 in order to be divisible by 18. So let's write that down. So we have to have
a 2 times 3 times 3. This makes it divisible by 18. If you multiply this
out, you actually get 18. And now let's look at the 12. So this part right over
here-- let me make it clear. This part right over
here is the part that makes up 18, makes
it divisible by 18. And then let's see. 12, we need two 2's and a 3. Well, we already have one 3,
so our 3 is taken care of. We have one 2, so this
2 is taken care of. But we don't have two 2s's. So we need another 2 here. So, notice, now this number
right over here has a 2 times 2 times 3 in it, or it has a
12 in it, and it has a 2 times 3 times 3, or an 18 in it. So this right over here is
the least common multiple of 18 and 12. If we multiply it out,
so 2 times 2 is 4. 4 times 3 is 12. 12 times 3 is equal to 36. And we are done. Now, the other way
you could've done it is what I would say
just the brute force method of just looking at the
multiples of these numbers. You would say, well, let's see. The multiples of 18
are 18, 36, and I could keep going
higher and higher, 54. And I could keep going. And the multiples of
12 are 12, 24, 36. And immediately I say, well, I
don't have to go any further. I already found a
multiple of both, and this is the smallest
multiple of both. It is 36. You might say, hey,
why would I ever do this one right over here
as opposed to this one? A couple of reasons. This one, you're
kind of-- it's fun, because you're actually
decomposing the number and then building it back up. And also, this is a better
way, especially if you're doing it with really, really
large and hairy numbers. Really, really, really
large and hairy numbers where you keep trying to
find all the multiples, you might have to go pretty
far to actually figure out what their least
common multiple is. Here, you'll be able to do it a
little bit more systematically, and you'll know
what you're doing.