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# Least common multiple: repeating factors

Video transcript

We need to figure out the least
common multiple of 30 and 25. So let's get our little
scratch pad out here. And we care about 30
and we care about 25. And I'm going to do this
using the prime factorization method which I just like more. Let's find the
prime factorization of both of these numbers. So 30, it's divisible by 2. It's 2 times 15. 15 is 3 times 5. And now we've expressed
30 as the product of only prime numbers,
2 times 3 times 5. Now let's do the
same thing for 25. 25 is-- well that's
just 5 times 5. So let me write that down. 25 is equal to 5 times 5. Now to find the least
common multiple, let me write this down,
the least common multiple of 30 and 25 is going to
have a number whose prime factorization is a super
set of both of these or has all of these numbers
in them as many times as we have in any one of these. So it's the least
common multiple. Well it has to be
divisible by 30. So it's going to need a
2 times a 3 times a 5. This is what makes
it divisible by 30. But it needs to also
be divisible by 25. And in order to be
divisible by 25, you need to have two 5s in
your prime factorization. Right now our prime
factorization only has one 5. So let's throw. So we have one 5
right over here. We need another 5. So let's throw another
5 right over here. So now this thing
clearly has a 25 in it. It's clearly divisible by 25. And this is the least
common multiple. I could have, if we just
wanted a common multiple, we could have thrown
more factors here and it would have definitely
been divisible by 30 or 25, but this has the bare
minimum of prime factors necessary to be
divisible by 30 and 25. If I got rid of
any one of these, I wouldn't be divisible
by both anymore. If I got rid of this 2, I
wouldn't be divisible by 30 anymore. If I got rid of one of the 5s,
I wouldn't be divisible by 25 anymore. So let's just multiply it out. This is essentially
the prime factorization of our least common multiple. And this is equal to 2 times 3
is 6, 6 times 5 is 30, 30 times 5 is equal to 150. And of course, we can
check our answer, 150. Check it, and we got it right.