Least common multiple: repeating factors

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.