9:59:59.000,9:59:59.000 Find the least common multiple of 9:59:59.000,9:59:59.000 15x, 20, and x squared plus 5x. 9:59:59.000,9:59:59.000 So when you're trying to find 9:59:59.000,9:59:59.000 the least common multiple of 9:59:59.000,9:59:59.000 a bunch of numbers you wanna 9:59:59.000,9:59:59.000 break them down into their 9:59:59.000,9:59:59.000 smallest constiguent parts. 9:59:59.000,9:59:59.000 And if you're dealing with regular 9:59:59.000,9:59:59.000 numbers, those smallest 9:59:59.000,9:59:59.000 constiguent parts are the 9:59:59.000,9:59:59.000 prime factorization numbers. 9:59:59.000,9:59:59.000 And if you're dealing with 9:59:59.000,9:59:59.000 things that have variable expressions 9:59:59.000,9:59:59.000 you just wanna factor them into 9:59:59.000,9:59:59.000 their simplest possible components. 9:59:59.000,9:59:59.000 You can't really call 9:59:59.000,9:59:59.000 those prime factorizations. 9:59:59.000,9:59:59.000 So let's try to do it. 9:59:59.000,9:59:59.000 And once you do that, 9:59:59.000,9:59:59.000 the least common multiple 9:59:59.000,9:59:59.000 just needs to be the 9:59:59.000,9:59:59.000 smallest number that can be 9:59:59.000,9:59:59.000 factored into those constiguents 9:59:59.000,9:59:59.000 and those constiguents in them. 9:59:59.000,9:59:59.000 So let's factor each of them. 9:59:59.000,9:59:59.000 So if I factor 15x, 9:59:59.000,9:59:59.000 this is the same things as 9:59:59.000,9:59:59.000 15 times x and 15 9:59:59.000,9:59:59.000 is three times five. 9:59:59.000,9:59:59.000 Both three and five are prime. 9:59:59.000,9:59:59.000 So we can factor this as 9:59:59.000,9:59:59.000 3 times 5 times x. 9:59:59.000,9:59:59.000 That's the, well, for the 9:59:59.000,9:59:59.000 coefficient we've done the 9:59:59.000,9:59:59.000 prime factorization. 9:59:59.000,9:59:59.000 And then x, 9:59:59.000,9:59:59.000 that's as much as we can factor 9:59:59.000,9:59:59.000 we don't know if x 9:59:59.000,9:59:59.000 is prime or not. 9:59:59.000,9:59:59.000 It's a variable. 9:59:59.000,9:59:59.000 Let's do the same thing for 20. 9:59:59.000,9:59:59.000 So 20 here 9:59:59.000,9:59:59.000 can be factored into 9:59:59.000,9:59:59.000 2 and 10. 9:59:59.000,9:59:59.000 And 10 can be factored 9:59:59.000,9:59:59.000 into 2 and 5. 9:59:59.000,9:59:59.000 So 20 is equal to 2 times 9:59:59.000,9:59:59.000 2 times 5. 9:59:59.000,9:59:59.000 And that's just a straight factorisation 9:59:59.000,9:59:59.000 Now let's do x square plus 5x 9:59:59.000,9:59:59.000 x squared plus 5x, we can factor out 9:59:59.000,9:59:59.000 and x, both of these terms are divisible by x 9:59:59.000,9:59:59.000 so this is equal to x times x plus 5 9:59:59.000,9:59:59.000 Right, if you divide... if you factor out an x here 9:59:59.000,9:59:59.000 you just get an x, if you factor out a x 9:59:59.000,9:59:59.000 out of 5x, you just get a 5. 9:59:59.000,9:59:59.000 And so the least common multiple, so let me write it down, 9:59:59.000,9:59:59.000 the least common multiple has to have the smallest number that has all of these components 9:59:59.000,9:59:59.000 at least, so it has all of these factors in it. 9:59:59.000,9:59:59.000 So let's just start with the smallest numbers and then we'll get to the variables. 9:59:59.000,9:59:59.000 So it has to have at least to 2s, because we have two 2s here, 9:59:59.000,9:59:59.000 we don't have any of them here, we have to have at least two 2s. 9:59:59.000,9:59:59.000 So we have two, let me do one of those pink 2s, 9:59:59.000,9:59:59.000 so it has to have at least two 2s, 9:59:59.000,9:59:59.000 two times two, two times two. 9:59:59.000,9:59:59.000 If it's going to be divisible by 20, it's also gonna need a 5, 9:59:59.000,9:59:59.000 but we'll get to that in a second. 9:59:59.000,9:59:59.000 So we have to have at least two 2s, and then it needs to have at least one three, 9:59:59.000,9:59:59.000 it needs at least one three, if it has any chance of being divisible by 15x. 9:59:59.000,9:59:59.000 So it has to have at least one 3, none of these guys, 9:59:59.000,9:59:59.000 so at least one 3, and then 5 9:59:59.000,9:59:59.000 if it has any chance of being divisible by 15x, it has to have 9:59:59.000,9:59:59.000 at least one 5, if it has any chance of being divisible by 20, 9:59:59.000,9:59:59.000 it has to have at least one 5, so it has to have at least one 5. 9:59:59.000,9:59:59.000 This one 5 right over here, will help make it sure that it is divisible 9:59:59.000,9:59:59.000 both by 15x and 20, although we haven't put all of the factors there. 9:59:59.000,9:59:59.000 Although this is already divisible by 20, because we have the 2 times 2 times 5 here 9:59:59.000,9:59:59.000 This isn't quite divisible by 15x yet, 9:59:59.000,9:59:59.000 because we don't have an x here. 9:59:59.000,9:59:59.000 It is divisible already by 15, 9:59:59.000,9:59:59.000 because we already have the 3 times 5. 9:59:59.000,9:59:59.000 And then you get to the x. This right over here has one x value, 9:59:59.000,9:59:59.000 so in order to be divisible by 15x, it has to have at least one x here 9:59:59.000,9:59:59.000 And so this is already divisible by 15x, you have 15x right over here 9:59:59.000,9:59:59.000 3 times 5 times x 9:59:59.000,9:59:59.000 It's already divisible by 20 9:59:59.000,9:59:59.000 you have 2 times 2 times 5, that is 20. 9:59:59.000,9:59:59.000 Is it divisible by x square + 5x? 9:59:59.000,9:59:59.000 Well it does have this x here, 9:59:59.000,9:59:59.000 but it still doesn't have an x plus 5 in it. 9:59:59.000,9:59:59.000 It still doesn't have and x plus 5, 9:59:59.000,9:59:59.000 so let me do that in orange. 9:59:59.000,9:59:59.000 So this, the least common multiple also needs, 9:59:59.000,9:59:59.000 an x plus 5 over here. 9:59:59.000,9:59:59.000 And so this is the least common multiple. 9:59:59.000,9:59:59.000 If you wanna multiply it out, to may be simplify it a little bit, 9:59:59.000,9:59:59.000 2 times 2 is 4, 4 times 3 is 12, 12 times 5 is 60, 9:59:59.000,9:59:59.000 60 times x is 60x, 9:59:59.000,9:59:59.000 so this is 60x 9:59:59.000,9:59:59.000 60x times x plus 5. 9:59:59.000,9:59:59.000 And if we want we can multiply it out. 9:59:59.000,9:59:59.000 60x times x plus 5 is 9:59:59.000,9:59:59.000 60x squared plus, 60 times 5 is 300, 300x 9:59:59.000,9:59:59.000 And there you go, the least common multiple.