Algebraic Expression Least Common Multiple

u15_l1_t3_we1 Algebraic Expression Least Common Multiple

Find the least common multiple of

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

So when you're trying to find

the least common multiple of

a bunch of numbers you wanna

break them down into their

smallest constiguent parts.

And if you're dealing with regular

numbers, those smallest

constiguent parts are the

prime factorization numbers.

And if you're dealing with

things that have variable expressions

you just wanna factor them into

their simplest possible components.

You can't really call

those prime factorizations.

So let's try to do it.

And once you do that,

the least common multiple

just needs to be the

smallest number that can be

factored into those constiguents

and those constiguents in them.

So let's factor each of them.

So if I factor 15x,

this is the same things as

15 times x and 15

is three times five.

Both three and five are prime.

So we can factor this as

3 times 5 times x.

That's the, well, for the

coefficient we've done the

prime factorization.

And then x,

that's as much as we can factor

we don't know if x

is prime or not.

It's a variable.

Let's do the same thing for 20.

So 20 here

can be factored into

2 and 10.

And 10 can be factored

into 2 and 5.

So 20 is equal to 2 times

2 times 5.

And that's just a straight factorisation

Now let's do x square plus 5x

x squared plus 5x, we can factor out

and x, both of these terms are divisible by x

so this is equal to x times x plus 5

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

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

out of 5x, you just get a 5.

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

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

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

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

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

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

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

so it has to have at least two 2s,

two times two, two times two.

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

but we'll get to that in a second.

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

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

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

so at least one 3, and then 5

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

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

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

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

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

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

This isn't quite divisible by 15x yet,

because we don't have an x here.

It is divisible already by 15,

because we already have the 3 times 5.

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

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

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

3 times 5 times x

It's already divisible by 20

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

Is it divisible by x square + 5x?

Well it does have this x here,

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

It still doesn't have and x plus 5,

so let me do that in orange.

So this, the least common multiple also needs,

an x plus 5 over here.

And so this is the least common multiple.

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

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

60 times x is 60x,

so this is 60x

60x times x plus 5.

And if we want we can multiply it out.

60x times x plus 5 is

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

And there you go, the least common multiple.