Find the LCM of six, 15, and 40.
Now before we attempt this question, we just need to know what LCM means. It means least common multiple or the lowest common multiple. Now let’s take each of those words one by one. So multiple, when you take a number and you multiply it by an integer, you get a multiple of that number. So six times one is the first multiple of six, six times two is the second multiple of six, six times three is the third multiple of six, and six times four is the fourth multiple of six. So the answers to these calculations — six, 12, 18, and 24 and so on — are all multiples of six.
Now the common means just what you think when things have got something in common. There for example, if I list out the multiples of three — that’s three, six, nine, and 12 — six is in both lists; so that’s a common multiple. And 12 is in both lists; so that’s a common multiple of six and three. So the common multiple is just a number that is a multiple of the six and the three.
That leaves us with least or lowest, so that just means the smallest of those common multiples. So in this case, six is the smallest of the numbers that are the common multiples; so it’s the least common multiple of three and six.
Now we’re gonna look at a couple of ways of doing this question. So the first one is just to write out all the multiples. So let’s write them out. One times 40 is 40, two times 40 is 80, three times 40 is 120, and so on. And the multiples of 15 are 15, 30, 45, 60, and so on. And the multiples of six are six, 12, 18, 24, and so on. So we’ve written out the multiples. We’ve now got to look for the smallest number that is in all three of those lists. So first of all, let’s make it clear that they are three different lists. And then the smallest number that I can find that isn’t in all three of those lists is 120. So writing out the multiples of all those numbers is one way of tackling this problem. But depending on what the numbers originally were, that may have been a pretty tricky task.
Now another way to tackle this problem is to find the prime factor decomposition of each of the numbers six, 15, and 40. So basically, we go through and try to divide by prime factors. So six, the smallest prime number is two. Is six divisible by two? Yes, it is, two times three. Well, two is a prime number and three is a prime number, so six is equal to two times three. And now 15, is 15 divisible by two? No, it’s not. So let’s try three. Yeah, that’s the next prime number and it is. And 15 is three times five, and three and five are both prime numbers. So 15 is equal to three times five. And then 40, is it divisible by two? Yes, it is, two times 20 and two is a prime number. 20, is that divisible by two? Yes, it is, two times 10 and two is a prime number. And what about 10, is that divisible by two? Yes, it is; it’s two times five. And two is a prime number, and five is a prime number. So we fully decomposed that into the product of some prime factors.
And now we can use those lists of prime factors of the three numbers in order to find- to calculate the lowest common multiple. So let’s start off with this one here. We’ve got a two as a factor of six. Well, 15 doesn’t have a factor of two, but 40 does have a factor of two. So we’re gonna tick those factors of two that we’ve found and say: the first factor of our lowest common multiple is two. And then we’re going to do the same again. So we’ve got another two, in fact here, in 40. But none of the other ones have got any more factors of two, so we’re just gonna write that one down once. And in fact, 40 has got another factor of two. So we’re gonna take that one off and write that down.
Now having dealt with all the twos, let’s try the three. So six has got a factor of three and 15 has got a factor of three. So let’s cross that one off. 40 doesn’t have a factor of three. So we write that down as a factor of our least common multiple. And now the only other factor we’ve got left is five. And 15 has got a factor of five and 40 has also got a factor of five, so we can use that in our least common multiple.
So now, we can see we’ve used up all of the factors of all of these different numbers. And our least common multiple, it’s gonna be two times two times two times three times five. Well, three times five is 15. If we double 15, we get 30. If we double 30, we get 60. And if we double 60, we get 120. So we get the same answer, either method. But the prime factor decomposition can turn out to be a quicker and easier method if you’ve got slightly trickier numbers to work with in the first place.