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Find the least common multiple of

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15x, 20, and x squared plus 5x.

3
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So when you're trying to find

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the least common multiple of

5
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a bunch of numbers you wanna

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break them down into their

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smallest constiguent parts.

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And if you're dealing with regular

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numbers, those smallest

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constiguent parts are the

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prime factorization numbers.

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And if you're dealing with

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things that have variable expressions

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you just wanna factor them into

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their simplest possible components.

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You can't really call

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those prime factorizations.

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So let's try to do it.

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And once you do that,

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the least common multiple

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just needs to be the

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smallest number that can be

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factored into those constiguents

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and those constiguents in them.

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So let's factor each of them.

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So if I factor 15x,

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this is the same things as

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15 times x and 15

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is three times five.

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Both three and five are prime.

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So we can factor this as

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3 times 5 times x.

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That's the, well, for the

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coefficient we've done the

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prime factorization.

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And then x,

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that's as much as we can factor

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we don't know if x

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is prime or not.

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It's a variable.

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Let's do the same thing for 20.

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So 20 here

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can be factored into

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2 and 10.

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And 10 can be factored

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into 2 and 5.

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So 20 is equal to 2 times

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2 times 5.

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And that's just a straight factorisation

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Now let's do x square plus 5x

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x squared plus 5x, we can factor out

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and x, both of these terms are divisible by x

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so this is equal to x times x plus 5

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Right, if you divide... if you factor out an x here

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you just get an x, if you factor out a x

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out of 5x, you just get a 5.

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And so the least common multiple, so let me write it down,

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the least common multiple has to have the smallest number that has all of these components

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at least, so it has all of these factors in it.

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So let's just start with the smallest numbers and then we'll get to the variables.

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So it has to have at least to 2s, because we have two 2s here,

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we don't have any of them here, we have to have at least two 2s.

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So we have two, let me do one of those pink 2s,

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so it has to have at least two 2s,

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two times two, two times two.

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If it's going to be divisible by 20, it's also gonna need a 5,

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but we'll get to that in a second.

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So we have to have at least two 2s, and then it needs to have at least one three,

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it needs at least one three, if it has any chance of being divisible by 15x.

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So it has to have at least one 3, none of these guys,

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so at least one 3, and then 5

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if it has any chance of being divisible by 15x, it has to have

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at least one 5, if it has any chance of being divisible by 20,

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it has to have at least one 5, so it has to have at least one 5.

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This one 5 right over here, will help make it sure that it is divisible

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both by 15x and 20, although we haven't put all of the factors there.

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Although this is already divisible by 20, because we have the 2 times 2 times 5 here

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This isn't quite divisible by 15x yet,

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because we don't have an x here.

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It is divisible already by 15,

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because we already have the 3 times 5.

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And then you get to the x. This right over here has one x value,

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so in order to be divisible by 15x, it has to have at least one x here

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And so this is already divisible by 15x, you have 15x right over here

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3 times 5 times x

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It's already divisible by 20

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you have 2 times 2 times 5, that is 20.

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Is it divisible by x square + 5x?

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Well it does have this x here,

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but it still doesn't have an x plus 5 in it.

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It still doesn't have and x plus 5,

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so let me do that in orange.

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So this, the least common multiple also needs,

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an x plus 5 over here.

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And so this is the least common multiple.

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If you wanna multiply it out, to may be simplify it a little bit,

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2 times 2 is 4, 4 times 3 is 12, 12 times 5 is 60,

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60 times x is 60x,

99
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so this is 60x

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60x times x plus 5.

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And if we want we can multiply it out.

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60x times x plus 5 is

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60x squared plus, 60 times 5 is 300, 300x

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And there you go, the least common multiple.