Least common multiple

what is the least common multiple of 36 and 12 so when they say another way to say this is LCM in parentheses 36 and 12 and this is literally saying what's the least common multiple of 36 and 12 well this one might pop out at you because 36 itself is a multiple of 12 and 36 is also a multiple of 36 it's 1 times 36 so the smallest number that is both a multiple of 36 and 12 because 36 is a multiple of 12 is actually 36 there we go let's do a couple more of these that one was too easy what is the least common multiple of 18 and 12 and they just state this with a different notation what's the least common multiple is common multiple of 18 and 12 is equal to question mark so let's think about this a little bit so there's a couple of ways you can think about so let's just write down our numbers that we care about we care about 18 and we care about 12 so there's two ways that we can approach this one is the prime factorization approach we can take the prime factorization of both of these numbers and then construct the smallest number whose prime factorization has all of the ingredients of both of these numbers and that will be the least common multiple so let's do that 18 is 2 times 9 which is the same thing as 2 times 3 times 3 or 18 is 2 times 9 9 is 3 times 3 so we could write 18 is equal to 2 times 3 times 3 that's its prime factorization 12 is 2 times 6 6 is 2 times 3 so 12 is equal to 2 times 2 times 3 now the least common multiple of 18 and 12 let me write this down so the least common multiple of 18 and 12 18 and 12 is going to have to have enough prime factors to cover both of these numbers and no more because we want the least common multiple or the smallest common multiple so let's think about it well it needs to have at least one 2 a 3 and a 3 in order to be divisible 18 so let's write that down so we have to have a 2 times 3 times 3 this makes it divisible by 18 if you multiply this out you actually get 18 and now let's look at the 12 so this part right over here let me make it clear this part right over here is the part that makes up 18 makes it divisible by 18 and then let's see 12 we need two 2s and a 3 well we already have we already have 1 3 so our 3 is taken care of we have 1 2 so this 2 is taken care of but we don't have two twos so we need another 2 here so notice now this number right over here has a 2 times 2 times 3 in it or it has a 12 in it and it has a 2 times 3 times 3 or an 18 in it so this right over here is the least common multiple of 18 and 12 if we multiply it out so 2 times 2 is 4 4 times 3 is 12 12 times 3 is equal to is equal to 36 and we are done now the other way you could have done it is what I would say just the brute force method of just looking at the multiples of these numbers you would say well let's see the multiples of 18 are 18 36 and I could keep going higher and higher 54 and I could keep going and the multiples of 12 are 12 24 36 and immediately I see what I don't have to go any I don't have to go any further I already found a multiple of both and this is the smallest multiple of both it is 36 you might say hey why would I ever do this one right over here as opposed to this one a couple of reasons this one you're kind of it's fun because you're actually decomposing the number and then building it back up and also this is a better way especially if you're doing really really large and hairy numbers really really really hard large and hairy numbers where you keep trying to find all the multiples you might have to go pretty far to actually figure out what their least common multiple is here you'll be able to do it a little bit more systematically and you'll know what you're doing