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What is the least common multiple, abbreviated as LCM, of 15, 6 and 10?

2
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So the LCM is exactly what the word is saying, it is the least common multiple of these numbers.

3
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And I know that probably did not help you much. But lets actually work trough this problem.

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So to do that, lets think of the different multiples of 15, 6 and 10.

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and then find the smallest multiple, the least multiple they have in common.

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So, lets find the multiples of 15. You have: 1 times 15 is 15, two times 15 is 30,

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then if you add 15 again you get 45, you add 15 again you get 60, you add 15 again,

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you get 75, you add 15 again, you get 90, you add 15 again you get 105.

9
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and if still none of these are common multiples with these guys over here

10
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then you may have to go further, but I will stop here for now.

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Now that's the multiples of 15 up through 105. Obviously, we keep going from there. Now lets do the multiples of 6.

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Let's do the multiples of 6: 1 times 6 is 6, two times 6 is 12, 3 times 6 is 18, 4 times 6 is 24,

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5 times 6 is 30, 6 times 6 is 36, 7 times 6 is 42, 8 times 6 is 48,

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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 to of them over here.

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We have 30 and we have a 30, we have a 60 and a 60. So the smallest LCM...

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...so if we only cared about the least common multiple of 15 and 6.

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We would say it is 30. Lets write that down as an intermediate: the LCM of 15 and 6. So the least common multiple,

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the smallest multiple that they have in common we see over here. 15 times 2 is 30 and 6 times 5 is 30.

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So this is definitely a common multiple and it is the smallest of all of their LCMs.

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60 is also a common multiple, but it is a bigger one. This is the least common multiple. So this is 30.

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We have not thought of the 10 yet. So lets bring the 10 in there. I think you see where this is going.

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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,

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and 30 is a common multiple of 15 and 6 and it is the smallest common multiple of all of them.

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So it is actually the fact that the LCM of 15, 6 and 10 is equal to 30.

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Now, this is one way to find the least common multiple. Literally, just find and look at the multiples of each of the numbers...

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and then see what the smallest multiple is they have in common.

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Another way to do that, is to look at the prime factorization of each of these numbers

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and the LCM is the number that has all the elements of the prime factorization of these and nothing else.

29
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So let me show you what I mean by that. So, you can do it this way or you can say that 15 is the

30
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same thing as 3 times 5 and that's it. That is its prime factorization, 15 is 3 times 5, since both 3 and 5 are prime numbers.

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We can say that 6 is the same thing as 2 times 3. That's it, that is its prime factorization, since both 2 and 3 are prime.

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And then we can say that 10 is the same thing as 2 times 5. Both two and 5 are prime, so we are done factoring it.

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So the LCM of 15, 6 and 10, just needs to have all of these prime factors.

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And what I mean is... to be clear, in order to be divisible by 15

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it has to have at least one 3 and at least one 5 in its prime factorization, so it needs to have one 3 and at least one 5.

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By having a 3 times 5 in its prime factorization that ensures that this number is divisible by 15.

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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 is all we want.

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We just need one 3. So one 2 and one 3. That is 2 times 3 and ensures we are divisible by 6. And let me make it clear, this right here is the 15.

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And then to make sure we are divisible by 10, we need to have at least one 2 and one 5. These two over here make sure we are divisible by 10.

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and so we have all of them, this 2 x 3 x 5 has all of the prime factors of either 10, 6 or 15, so it is the LCM.

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So if we multiply this out, you will get, 2 x 3 is 6, 6 x 5 is 30.

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So either way. Hopefully these kind of resonate with you and you see why they make sense.

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This second way is a little bit better, if you are trying to do it for really complex numbers...

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...numbers, where you might have to be multiplying for a really long time.

45
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Well, either way, both of these are valid ways of finding the least common multiple.