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Least common multiple of three numbers

CCSS.Math:

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 15 6 and 10 and then find 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 you get 75 you had 15 again you get 90 you had 15 again you get 105 and if we 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 that's the multiples of 15 up through 105 obviously we can we can keep going from there we can keep going from there now let's do the multiples of 6 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 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 a kind of 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 multiple 60 is also a common multiple but it's a bigger one this is the least common multiple so this is 30 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 there 10 20 30 40 what we already went far enough because we already got to 30 and 30 is a common multiple 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 find 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 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 & 5 are prime we can say that 6 is the same thing as 2 times 3 that's it that's its prime factorization that both 2 & 3 are prime and then we can say that we can say that 10 is the same thing as 2 times 5 that's it both 2 & 5 are prime so we're done factoring it and so the least common multiple the least common multiple of 15 6 and 10 just needs to have all of these prime factors and what I mean that we clear is is in order to be divisible by 15 it has to have at least one and one three and one five in its prime factorization so I just have at least one three and at least one five by having it three times five in its prime factorization that ensures that this number is divisible by 15 to be divisible by six it has to have at least one two and one three so it has to have at least one two and we already have a three over here so that's all we want we just need one three so 1 2 & 1 3 this 2 times 3 ensures that we are divisible by 6 and let me make it here this is right here is the 15 and then to make sure that we're divisible by 10 we have to have at least one two and one five we have to have at least one two and one five 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 kind of resonate with you and you see why they make sense this second way is a little bit is a little bit better if you're trying to do it for kind of really complex numbers numbers where you might have to be multiplying it for a long time multiple well either way both of these are kind of valid ways of finding the least common multiple