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Dialogue: 0,0:00:00.37,0:00:07.60,Default,,0000,0000,0000,,What is the least common multiple, abbreviated as LCM, of 15, 6 and 10?
Dialogue: 0,0:00:07.60,0:00:13.98,Default,,0000,0000,0000,,So the LCM is exactly what the word is saying, it is the least common multiple of these numbers.
Dialogue: 0,0:00:13.98,0:00:17.45,Default,,0000,0000,0000,,And I know that probably did not help you much. But lets actually work trough this problem.
Dialogue: 0,0:00:17.45,0:00:22.28,Default,,0000,0000,0000,,So to do that, lets think of the different multiples of 15, 6 and 10.
Dialogue: 0,0:00:22.28,0:00:26.45,Default,,0000,0000,0000,,and then find the smallest multiple, the least multiple they have in common.
Dialogue: 0,0:00:26.45,0:00:34.40,Default,,0000,0000,0000,,So, lets find the multiples of 15. You have: 1 times 15 is 15, two times 15 is 30,
Dialogue: 0,0:00:34.40,0:00:41.37,Default,,0000,0000,0000,,then if you add 15 again you get 45, you add 15 again you get 60, you add 15 again,
Dialogue: 0,0:00:41.37,0:00:49.01,Default,,0000,0000,0000,,you get 75, you add 15 again, you get 90, you add 15 again you get 105.
Dialogue: 0,0:00:49.01,0:00:53.81,Default,,0000,0000,0000,,and if still none of these are common multiples with these guys over here
Dialogue: 0,0:00:54.10,0:00:56.91,Default,,0000,0000,0000,,then you may have to go further, but I will stop here for now.
Dialogue: 0,0:00:57.09,0:01:07.12,Default,,0000,0000,0000,,Now that's the multiples of 15 up through 105. Obviously, we keep going from there. Now lets do the multiples of 6.
Dialogue: 0,0:01:07.12,0:01:17.48,Default,,0000,0000,0000,,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,
Dialogue: 0,0:01:17.48,0:01:27.34,Default,,0000,0000,0000,,5 times 6 is 30, 6 times 6 is 36, 7 times 6 is 42, 8 times 6 is 48,
Dialogue: 0,0:01:27.34,0:01:39.73,Default,,0000,0000,0000,,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.
Dialogue: 0,0:01:39.73,0:01:44.68,Default,,0000,0000,0000,,We have 30 and we have a 30, we have a 60 and a 60. So the smallest LCM...
Dialogue: 0,0:01:44.68,0:01:47.69,Default,,0000,0000,0000,,...so if we only cared about the least common multiple of 15 and 6.
Dialogue: 0,0:01:47.80,0:01:57.36,Default,,0000,0000,0000,,We would say it is 30. Lets write that down as an intermediate: the LCM of 15 and 6. So the least common multiple,
Dialogue: 0,0:01:57.36,0:02:06.53,Default,,0000,0000,0000,,the smallest multiple that they have in common we see over here. 15 times 2 is 30 and 6 times 5 is 30.
Dialogue: 0,0:02:06.60,0:02:10.80,Default,,0000,0000,0000,,So this is definitely a common multiple and it is the smallest of all of their LCMs.
Dialogue: 0,0:02:10.90,0:02:16.32,Default,,0000,0000,0000,,60 is also a common multiple, but it is a bigger one. This is the least common multiple. So this is 30.
Dialogue: 0,0:02:16.62,0:02:22.86,Default,,0000,0000,0000,,We have not thought of the 10 yet. So lets bring the 10 in there. I think you see where this is going.
Dialogue: 0,0:02:22.92,0:02:30.59,Default,,0000,0000,0000,,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,
Dialogue: 0,0:02:30.59,0:02:38.97,Default,,0000,0000,0000,,and 30 is a common multiple of 15 and 6 and it is the smallest common multiple of all of them.
Dialogue: 0,0:02:39.16,0:02:47.41,Default,,0000,0000,0000,,So it is actually the fact that the LCM of 15, 6 and 10 is equal to 30.
Dialogue: 0,0:02:47.49,0:02:52.92,Default,,0000,0000,0000,,Now, this is one way to find the least common multiple. Literally, just find and look at the multiples of each of the numbers...
Dialogue: 0,0:02:52.98,0:02:57.33,Default,,0000,0000,0000,,and then see what the smallest multiple is they have in common.
Dialogue: 0,0:02:57.33,0:03:01.97,Default,,0000,0000,0000,,Another way to do that, is to look at the prime factorization of each of these numbers
Dialogue: 0,0:03:02.04,0:03:08.66,Default,,0000,0000,0000,,and the LCM is the number that has all the elements of the prime factorization of these and nothing else.
Dialogue: 0,0:03:08.75,0:03:14.42,Default,,0000,0000,0000,,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
Dialogue: 0,0:03:14.42,0:03:23.54,Default,,0000,0000,0000,,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.
Dialogue: 0,0:03:23.61,0:03:30.78,Default,,0000,0000,0000,,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.
Dialogue: 0,0:03:30.78,0:03:40.25,Default,,0000,0000,0000,,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.
Dialogue: 0,0:03:40.25,0:03:50.93,Default,,0000,0000,0000,,So the LCM of 15, 6 and 10, just needs to have all of these prime factors.
Dialogue: 0,0:03:50.93,0:03:55.60,Default,,0000,0000,0000,,And what I mean is... to be clear, in order to be divisible by 15
Dialogue: 0,0:03:55.60,0:04:03.67,Default,,0000,0000,0000,,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.
Dialogue: 0,0:04:03.76,0:04:09.60,Default,,0000,0000,0000,,By having a 3 times 5 in its prime factorization that ensures that this number is divisible by 15.
Dialogue: 0,0:04:09.66,0:04:18.45,Default,,0000,0000,0000,,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.
Dialogue: 0,0:04:18.57,0:04:28.35,Default,,0000,0000,0000,,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.
Dialogue: 0,0:04:28.95,0:04:41.88,Default,,0000,0000,0000,,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.
Dialogue: 0,0:04:42.08,0:04:47.66,Default,,0000,0000,0000,,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.
Dialogue: 0,0:04:52.92,0:04:52.92,Default,,0000,0000,0000,,So if we multiply this out, you will get, 2 x 3 is 6, 6 x 5 is 30.
Dialogue: 0,0:04:55.97,0:05:05.47,Default,,0000,0000,0000,,So either way. Hopefully these kind of resonate with you and you see why they make sense.
Dialogue: 0,0:05:05.59,0:05:13.19,Default,,0000,0000,0000,,This second way is a little bit better, if you are trying to do it for really complex numbers...
Dialogue: 0,0:05:13.19,0:05:16.06,Default,,0000,0000,0000,,...numbers, where you might have to be multiplying for a really long time.
Dialogue: 0,0:05:16.06,0:05:21.83,Default,,0000,0000,0000,,Well, either way, both of these are valid ways of finding the least common multiple.