WEBVTT 99:59:59.999 --> 99:59:59.999 Find the least common multiple of 99:59:59.999 --> 99:59:59.999 15x, 20, and x squared plus 5x. 99:59:59.999 --> 99:59:59.999 So when you're trying to find 99:59:59.999 --> 99:59:59.999 the least common multiple of 99:59:59.999 --> 99:59:59.999 a bunch of numbers you wanna 99:59:59.999 --> 99:59:59.999 break them down into their 99:59:59.999 --> 99:59:59.999 smallest constiguent parts. 99:59:59.999 --> 99:59:59.999 And if you're dealing with regular 99:59:59.999 --> 99:59:59.999 numbers, those smallest 99:59:59.999 --> 99:59:59.999 constiguent parts are the 99:59:59.999 --> 99:59:59.999 prime factorization numbers. 99:59:59.999 --> 99:59:59.999 And if you're dealing with 99:59:59.999 --> 99:59:59.999 things that have variable expressions 99:59:59.999 --> 99:59:59.999 you just wanna factor them into 99:59:59.999 --> 99:59:59.999 their simplest possible components. 99:59:59.999 --> 99:59:59.999 You can't really call 99:59:59.999 --> 99:59:59.999 those prime factorizations. 99:59:59.999 --> 99:59:59.999 So let's try to do it. 99:59:59.999 --> 99:59:59.999 And once you do that, 99:59:59.999 --> 99:59:59.999 the least common multiple 99:59:59.999 --> 99:59:59.999 just needs to be the 99:59:59.999 --> 99:59:59.999 smallest number that can be 99:59:59.999 --> 99:59:59.999 factored into those constiguents 99:59:59.999 --> 99:59:59.999 and those constiguents in them. 99:59:59.999 --> 99:59:59.999 So let's factor each of them. 99:59:59.999 --> 99:59:59.999 So if I factor 15x, 99:59:59.999 --> 99:59:59.999 this is the same things as 99:59:59.999 --> 99:59:59.999 15 times x and 15 99:59:59.999 --> 99:59:59.999 is three times five. 99:59:59.999 --> 99:59:59.999 Both three and five are prime. 99:59:59.999 --> 99:59:59.999 So we can factor this as 99:59:59.999 --> 99:59:59.999 3 times 5 times x. 99:59:59.999 --> 99:59:59.999 That's the, well, for the 99:59:59.999 --> 99:59:59.999 coefficient we've done the 99:59:59.999 --> 99:59:59.999 prime factorization. 99:59:59.999 --> 99:59:59.999 And then x, 99:59:59.999 --> 99:59:59.999 that's as much as we can factor 99:59:59.999 --> 99:59:59.999 we don't know if x 99:59:59.999 --> 99:59:59.999 is prime or not. 99:59:59.999 --> 99:59:59.999 It's a variable. 99:59:59.999 --> 99:59:59.999 Let's do the same thing for 20. 99:59:59.999 --> 99:59:59.999 So 20 here 99:59:59.999 --> 99:59:59.999 can be factored into 99:59:59.999 --> 99:59:59.999 2 and 10. 99:59:59.999 --> 99:59:59.999 And 10 can be factored 99:59:59.999 --> 99:59:59.999 into 2 and 5. 99:59:59.999 --> 99:59:59.999 So 20 is equal to 2 times 99:59:59.999 --> 99:59:59.999 2 times 5. 99:59:59.999 --> 99:59:59.999 And that's just a straight factorisation 99:59:59.999 --> 99:59:59.999 Now let's do x square plus 5x 99:59:59.999 --> 99:59:59.999 x squared plus 5x, we can factor out 99:59:59.999 --> 99:59:59.999 and x, both of these terms are divisible by x 99:59:59.999 --> 99:59:59.999 so this is equal to x times x plus 5 99:59:59.999 --> 99:59:59.999 Right, if you divide... if you factor out an x here 99:59:59.999 --> 99:59:59.999 you just get an x, if you factor out a x 99:59:59.999 --> 99:59:59.999 out of 5x, you just get a 5. 99:59:59.999 --> 99:59:59.999 And so the least common multiple, so let me write it down, 99:59:59.999 --> 99:59:59.999 the least common multiple has to have the smallest number that has all of these components 99:59:59.999 --> 99:59:59.999 at least, so it has all of these factors in it. 99:59:59.999 --> 99:59:59.999 So let's just start with the smallest numbers and then we'll get to the variables. 99:59:59.999 --> 99:59:59.999 So it has to have at least to 2s, because we have two 2s here, 99:59:59.999 --> 99:59:59.999 we don't have any of them here, we have to have at least two 2s. 99:59:59.999 --> 99:59:59.999 So we have two, let me do one of those pink 2s, 99:59:59.999 --> 99:59:59.999 so it has to have at least two 2s, 99:59:59.999 --> 99:59:59.999 two times two, two times two. 99:59:59.999 --> 99:59:59.999 If it's going to be divisible by 20, it's also gonna need a 5, 99:59:59.999 --> 99:59:59.999 but we'll get to that in a second. 99:59:59.999 --> 99:59:59.999 So we have to have at least two 2s, and then it needs to have at least one three, 99:59:59.999 --> 99:59:59.999 it needs at least one three, if it has any chance of being divisible by 15x. 99:59:59.999 --> 99:59:59.999 So it has to have at least one 3, none of these guys, 99:59:59.999 --> 99:59:59.999 so at least one 3, and then 5 99:59:59.999 --> 99:59:59.999 if it has any chance of being divisible by 15x, it has to have 99:59:59.999 --> 99:59:59.999 at least one 5, if it has any chance of being divisible by 20, 99:59:59.999 --> 99:59:59.999 it has to have at least one 5, so it has to have at least one 5. 99:59:59.999 --> 99:59:59.999 This one 5 right over here, will help make it sure that it is divisible 99:59:59.999 --> 99:59:59.999 both by 15x and 20, although we haven't put all of the factors there. 99:59:59.999 --> 99:59:59.999 Although this is already divisible by 20, because we have the 2 times 2 times 5 here 99:59:59.999 --> 99:59:59.999 This isn't quite divisible by 15x yet, 99:59:59.999 --> 99:59:59.999 because we don't have an x here. 99:59:59.999 --> 99:59:59.999 It is divisible already by 15, 99:59:59.999 --> 99:59:59.999 because we already have the 3 times 5. 99:59:59.999 --> 99:59:59.999 And then you get to the x. This right over here has one x value, 99:59:59.999 --> 99:59:59.999 so in order to be divisible by 15x, it has to have at least one x here 99:59:59.999 --> 99:59:59.999 And so this is already divisible by 15x, you have 15x right over here 99:59:59.999 --> 99:59:59.999 3 times 5 times x 99:59:59.999 --> 99:59:59.999 It's already divisible by 20 99:59:59.999 --> 99:59:59.999 you have 2 times 2 times 5, that is 20. 99:59:59.999 --> 99:59:59.999 Is it divisible by x square + 5x? 99:59:59.999 --> 99:59:59.999 Well it does have this x here, 99:59:59.999 --> 99:59:59.999 but it still doesn't have an x plus 5 in it. 99:59:59.999 --> 99:59:59.999 It still doesn't have and x plus 5, 99:59:59.999 --> 99:59:59.999 so let me do that in orange. 99:59:59.999 --> 99:59:59.999 So this, the least common multiple also needs, 99:59:59.999 --> 99:59:59.999 an x plus 5 over here. 99:59:59.999 --> 99:59:59.999 And so this is the least common multiple. 99:59:59.999 --> 99:59:59.999 If you wanna multiply it out, to may be simplify it a little bit, 99:59:59.999 --> 99:59:59.999 2 times 2 is 4, 4 times 3 is 12, 12 times 5 is 60, 99:59:59.999 --> 99:59:59.999 60 times x is 60x, 99:59:59.999 --> 99:59:59.999 so this is 60x 99:59:59.999 --> 99:59:59.999 60x times x plus 5. 99:59:59.999 --> 99:59:59.999 And if we want we can multiply it out. 99:59:59.999 --> 99:59:59.999 60x times x plus 5 is 99:59:59.999 --> 99:59:59.999 60x squared plus, 60 times 5 is 300, 300x 99:59:59.999 --> 99:59:59.999 And there you go, the least common multiple.