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All right.
So now that we know what multiples are,
let's look at two or more terms, and look for common multiples.
So as we showed before, when you have a integer or polynomial,
there can be several multiples.
Infinitely many, in fact.
Right?
So if you have two or more integers or polynomials,
you can have multiples of them.
But what we want to know is, can we have a common multiple?
So let's take a look at what that means.
Let's look at the numbers 2, 3, 4, and 6.
And I say, find multiples of 2.
Well, all multiples of 2 are 2, 4, 6, 8, 10, 12, and so on.
What are multiples of 3?
3, 6-- can you see?
9, 12, and so on.
Multiples of 4 are 4, 8, 12.
Multiples of 6 are 6, 12, and so on.
So all of these have their multiples.
The question is, can we find a multiple of all four of them
at once?
So can you find a number where that number is a multiple of 2,
it's a multiple of 3, multiple of 4, and a multiple of 6?
And we're saying 24 is such number because 24
is 2 times 12, or I can write it as 3 times what?
Good, 8.
I can also write it as 4 times 6, good.
I can also write it as 6 times 4.
So 24 can be written as these numbers times something,
so it's a multiple of all of them.
So 24 is a common multiple of 2, 3, 4, and 6.
All right.
Let's look at a polynomial.
We're saying that this quantity here is a multiple of x.
Well, yes, because it just needs x to the power of 1,
and it has x to the power of 31, so that matches up.
This number is also a multiple of this polynomial
here because we need x to the third.
So 3 or more power of x, got it.
3x plus 1, 5 or more power, got it.
x plus square root of 3, 12 or more power, got it.
So this is also a multiple of that quantity.
Let's look at here.
x squared [INAUDIBLE] to the x power is way more.
And 3x plus 1, power is more, so yes.
And this one also, x plus square root 3 to the power of 24,
and here we really have 2.
So this polynomial here is a multiple of x.
It's a multiple of x cubed times 3x plus 1 to the fifth power,
times x plus square root 3 to the 12th power.
It's a multiple of that polynomial,
and that polynomial.
So it's a common multiple.
There are infinitely many common multiples
of two or more integers or polynomials.
The smallest amongst all the common multiples
is called the least common multiple.
Now when you're working with variables,
what does it mean to say smallest common multiple?
All it means is that all the factors are there.
And the smallest exponents that we can have,
that's what will be called least common multiple.
So for example, 24 is a common multiple of 2, 3, 4, and 6.
But 12 is the least common multiple of 2,
3, 4, 6 because if you looked at all multiples of 2,
3, 4, 6, that would be 12, 24, 36, 48, and so on.
And 12 is the smallest such number.
All right.
Let's take a look at this polynomial.
This polynomial is going to be the least common multiple
of these polynomials.
Why is that?
Because I'm taking the smallest power of x that's needed.
See, we have x to the third, x squared, no x here.
So we need an x, but x to the third is sufficient.
3x plus 1, we need to the power, largest power [? that ?]
up here, so that would be the 5.
And x plus square root 3 to the power of 12.
So those three things here makes it a common multiple,
but in fact, a least common multiple.
If you reduce the power of any of these factors,
it will no longer be a common multiple of the 3 polynomials
that are listed here.
All right.
So to find the least common multiple,
we have to factor everything because if you're don't factor,
then we can't really see least common multiple-- at least,
not easily.
So we have to find a way to write numbers and polynomials
as product of primes.
And so that's going to be what we're going to focus on.
So how can we write, given an object, into its primes
so that we can look for common multiples or least
common multiples and so on?
All right.
So let's just look at common multiples-- in fact,
least common multiples-- and how to find them.
We're going to do a few examples.
Take good notes because eventually in [? real logs, ?]
you'll have to do that yourself.
All right.
So here are two numbers.
One number, second number.
We're saying that this number here
is the least common multiple of those two.
Why?
All right.
So let's see what we need.
We need all prime factors that appear in both numbers.
So 2 is there, 3 is there, 7 is there.
So the first number is taken care of.
All the prime factors are there.
What about the second number?
2 is there, 3 is there, 5 is there.
Good.
So all prime numbers are there.
Now we just have to make sure that the prime factors have
the correct exponents on them.
So let's take a look at exponent.
We have 2 the 4 and 2 the 12th.
2 the 12th is the highest power, so we
need at least 12 2's for common multiples.
If you want least common multiple,
you're going to have to take 12.
All right.
Let's take our next.
3 to the 10th and 3 to the 4.
So 3 to the 10th is going to be the highest power.
All right.
Let's take a look at 5.
5 doesn't appear here, just appears here.
So 5 to the second is the highest power, so we need that.
And 7 doesn't appear here, just here.
So 7 to the third, we need third.
So this is the least common multiple
of these two quantities.
All right.
Here's the next example.
You want to find the least common multiple of these two
polynomials.
Well, the answer will show in just a little bit.
So pause the video here, do the problem,
and then check the answer.
Assuming you have come back from the break by pausing the video
and attempting it, let's see.
So we need, what?
We need x plus one, 3x minus 1, 1 minus x,
and x plus square root 3.
All of those factors have to be here.
Now we have to match the powers.
x to the 4, x to the 12, so we need 12 power.
Let's see, 3x minus 1, 10 power, 4 power, we need 10 power.
And then, what do we have here?
1 minus x squared.
There is no 1 minus x here, so we're
going to take that 2 power.
x plus square root three here, but not here,
so we're going to have to take this 3 power.
OK?
So what we've done is taken between x plus 1 to the 4 and x
plus 1 to the 12th, we need x plus 1 to the 12th power.
3x minus 1 to the 10 power, 3x minus 1 to the 4 power.
We need the 10 power.
These items, the next two factors
do not appear in the other term, so we
have to take the whole as they are.
So this is the least common multiple.
If you reduced the power of any of these factors by even 1,
it will no longer be a common multiple
of either the first or the second quantity.
Let's take some other examples.
So for the two polynomials that you see here-- a squared,
b to the 10, c cubed, b cubed, c to the 12, d to the fifth--
find the least common multiple.
I'll give you a few moments.
Go ahead and do that on your own.
All right.
Let's take a look.
We clearly need an a squared.
The other quantity doesn't even have any a in it,
so we're going to need all of a squared.
OK.
So that we need.
We need b to the 10, b to the third.
Which one should we pick, this one or this one?
Clearly that one because that's the higher power of b.
OK.
c to the 12 and c to the third, which would should we pick?
Well, c to the 12 has the higher power,
so we're going to have to [? need ?] that.
This one is all done.
d we haven't done yet. d to the fifth is here, not here.
So d to the fifth is going to have to [INAUDIBLE].
All right. d to the fifth is not here,
so we need all of d to the fifth.
So the circled items are going to be multiplied together
to give you your least common multiple.
So the least common multiple is going to be a to the second,
b to the 10, c to the 12, d to the fifth multiplied together.
All right.
Try that on your own.
Pause the video, please.
Do really do it because soon you're
going to have to do video log questions on it.
Remember, every factor that appears here and here
should be there.
If they are common factors, then you
must contain the highest power, highest exponent.
Let's take a look then.
It doesn't matter what order you go in, either.
So if we take here 5x plus 1 to the 12th power, 5x
plus 1 to the second power, which one do we need to keep?
It's that one because that has the higher power.
All right, let's see.
3 squared-- there is no 3 squared here,
so that will have to stay.
x minus 1 squared-- [? no x ?] minus 1,
so it will have to stay.
5x plus 1, we already took care of.
Let's see.
x plus square root 3 to the 7, x plus square root 3 to the 1.
So the 7 power is the highest power, so we have to keep that.
So all the factors in the first quantity are taken care of.
Let's see, did we take care of x cubed yet?
Nope, so x cubed will have to stay.
And x plus 1 to the 9 will have to stay also
because that was not anywhere else.
All right.
So 3 squared, x cubed, x minus 1 squared,
x plus 1 to the 9 power, 5x plus 1 to the 12th power,
and x plus square root 3 to the 7 power, multiplied
together will be the least common multiple.
All right.
Try that on your own.
Find the least common multiple of those three polynomials.
Pause the video.
Try it, please.
It's OK if you go wrong.
Wrong answers teach you things about your own thinking.
And then when you see the right answer,
you can fix the understanding if it is missing something.
All right.
Same thing again.
So look for x minus 1's.
Are there any x minus 1's?
Yes, but they both have one power, so we need to keep one.
x plus 1's, we have one power only,
so we need to keep one of those.
2x minus 3, there's only one of those, so keep one of those.
So our least common multiple is going to be x minus 1,
x plus 1, and 2x minus 3.
You will see examples like these when
we do addition of rational expressions quite a bit.
All right.
Let's take a look at common multiples of these numbers.
When you factor everything, then find least common multiples.
So remember what we said.
Write the numbers down, and then do what?
Good.
Start circling prime numbers that we need.
So we need 3 squared, 3 12, and 3.
Which one do we need to keep?
Clearly the higher power, which is the 12 here.
Can you see?
3 to the 12 is the highest power of 3,
so we're going to keep that.
OK.
So 3 is taken care of.
Let's start with 5.
5 to sixth, to the third, to the third.
So 5 to the 6, we're going to have to keep.
How about 7's?
7 to the 8, 7 to the 12, 7 to the second.
Which one do we need to keep?
12 because that's the highest power.
All right.
What else?
This number's all taken care of.
7 we took care-- 11.
11 to the third.
There is no other 11, so that will have to stay also.
So our least common multiple would be 3 the 12th,
5 to the 6, 7 to the 12th, 11 to the third.
That's a pretty huge number, isn't it?
If you actually computed it.
So think about what common multiple really is.
And if I asked you to find least common multiple of x plus 1
and x, notice that x alone, or x plus 1 alone,
cannot be it because x is an additive factor in x plus 1.
So the common multiple of x plus 1 and x
would have to be x times x plus 1., so that each x and x plus 1
are factors of it.