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.