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 so 30 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 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 of 30 30 and 25 is going to have a number whose prime factorization is a superset 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 least common multiple well it has to be divisible by 30 so it's going to need a 2 times the 3 times a 5 this is what makes it divisible by 30 but it needs to also be divisible by 25 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 1 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 had if we just wanted a common multiple we could have threw more we could have thrown more factors here and it would have definitely been divisible by 30 or 25 but it 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 any more 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