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Least common multiple of three numbers

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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 the 15, 6, and 10 and then find at 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 75. You add 15 again, you get 90. You add 15 again, you get 105. And 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 the multiples of 15 up through 105. Obviously, we can keep going from there. Now, 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 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 an 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 multiples. 60 is also a common multiple. But it's a bigger one. This is the least common multiple. So this is 30. Well, 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. They are 10, 20, 30, 40. Well, we already went far enough, because we already got to 30. And 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 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 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 and 5 are prime. We can say that 6 is the same thing as 2 times 3. That's it. That's its prime factorization. Both 2 and 3 are prime. And then we can say that 10 is the same thing as 2 times 5. Both 2 and 5 are prime. So we're done factoring it. And so the least common multiple of 15, 6, and 10 just needs to have all of these prime factors. And what I mean, to be clear, is in order to be divisible by 15, it has to have at least one 3 and one 5 in its prime factorization. So it has to have at least one 3 and at least one 5. By having a 3 times 5 in its prime factorization, that ensures that this number is divisible by 15. To be divisible by 6, it has to have at least one 2 and one 3. So it has to have at least one 2. And we already have a 3 over here. So that's all we want. We just need one 3. So one 2 and one 3, this 2 times 3, ensures that we are divisible by 6. And let me make it clear. This right here is the 15. And then to make sure that we're divisible by 10, we have to have at least one 2 and one 5. 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 resonate with you. And you see why they make sense. This second way is a little bit better if you're trying to do it for really complex numbers, numbers where you might have to be multiplying it for a long time. Either way, both of these are valid ways of finding the least common multiple.