WEBVTT 00:00:00.369 --> 00:00:07.602 What is the least common multiple, abbreviated as LCM, of 15, 6 and 10? 00:00:07.602 --> 00:00:13.984 So the LCM is exactly what the word is saying, it is the least common multiple of these numbers. 00:00:13.984 --> 00:00:17.453 And I know that probably did not help you much. But lets actually work trough this problem. 00:00:17.453 --> 00:00:22.275 So to do that, lets think of the different multiples of 15, 6 and 10. 00:00:22.275 --> 00:00:26.453 and then find the smallest multiple, the least multiple they have in common. 00:00:26.453 --> 00:00:34.396 So, lets find the multiples of 15. You have: 1 times 15 is 15, two times 15 is 30, 00:00:34.396 --> 00:00:41.373 then if you add 15 again you get 45, you add 15 again you get 60, you add 15 again, 00:00:41.373 --> 00:00:49.012 you get 75, you add 15 again, you get 90, you add 15 again you get 105. 00:00:49.012 --> 00:00:53.807 and if still none of these are common multiples with these guys over here 00:00:54.098 --> 00:00:56.906 then you may have to go further, but I will stop here for now. 00:00:57.090 --> 00:01:07.119 Now that's the multiples of 15 up through 105. Obviously, we keep going from there. Now lets do the multiples of 6. 00:01:07.119 --> 00:01:17.480 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, 00:01:17.480 --> 00:01:27.345 5 times 6 is 30, 6 times 6 is 36, 7 times 6 is 42, 8 times 6 is 48, 00:01:27.345 --> 00:01:39.734 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. 00:01:39.734 --> 00:01:44.684 We have 30 and we have a 30, we have a 60 and a 60. So the smallest LCM... 00:01:44.684 --> 00:01:47.689 ...so if we only cared about the least common multiple of 15 and 6. 00:01:47.797 --> 00:01:57.356 We would say it is 30. Lets write that down as an intermediate: the LCM of 15 and 6. So the least common multiple, 00:01:57.356 --> 00:02:06.526 the smallest multiple that they have in common we see over here. 15 times 2 is 30 and 6 times 5 is 30. 00:02:06.605 --> 00:02:10.803 So this is definitely a common multiple and it is the smallest of all of their LCMs. 00:02:10.896 --> 00:02:16.325 60 is also a common multiple, but it is a bigger one. This is the least common multiple. So this is 30. 00:02:16.617 --> 00:02:22.862 We have not thought of the 10 yet. So lets bring the 10 in there. I think you see where this is going. 00:02:22.923 --> 00:02:30.592 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, 00:02:30.592 --> 00:02:38.973 and 30 is a common multiple of 15 and 6 and it is the smallest common multiple of all of them. 00:02:39.158 --> 00:02:47.412 So it is actually the fact that the LCM of 15, 6 and 10 is equal to 30. 00:02:47.489 --> 00:02:52.920 Now, this is one way to find the least common multiple. Literally, just find and look at the multiples of each of the numbers... 00:02:52.982 --> 00:02:57.333 and then see what the smallest multiple is they have in common. 00:02:57.333 --> 00:03:01.973 Another way to do that, is to look at the prime factorization of each of these numbers 00:03:02.044 --> 00:03:08.658 and the LCM is the number that has all the elements of the prime factorization of these and nothing else. 00:03:08.750 --> 00:03:14.422 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 00:03:14.422 --> 00:03:23.537 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. 00:03:23.614 --> 00:03:30.783 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. 00:03:30.783 --> 00:03:40.249 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. 00:03:40.249 --> 00:03:50.930 So the LCM of 15, 6 and 10, just needs to have all of these prime factors. 00:03:50.930 --> 00:03:55.599 And what I mean is... to be clear, in order to be divisible by 15 00:03:55.599 --> 00:04:03.672 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. 00:04:03.765 --> 00:04:09.599 By having a 3 times 5 in its prime factorization that ensures that this number is divisible by 15. 00:04:09.661 --> 00:04:18.451 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. 00:04:18.574 --> 00:04:28.346 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. 00:04:28.946 --> 00:04:41.884 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. 00:04:42.083 --> 00:04:47.655 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. 00:04:52.922 --> 00:04:52.923 So if we multiply this out, you will get, 2 x 3 is 6, 6 x 5 is 30. 00:04:55.969 --> 00:05:05.471 So either way. Hopefully these kind of resonate with you and you see why they make sense. 00:05:05.594 --> 00:05:13.193 This second way is a little bit better, if you are trying to do it for really complex numbers... 00:05:13.193 --> 00:05:16.062 ...numbers, where you might have to be multiplying for a really long time. 00:05:16.062 --> 00:05:21.834 Well, either way, both of these are valid ways of finding the least common multiple.