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### Course: Algebra (all content) > Unit 13

Lesson 4: Adding & subtracting rational expressions- Adding & subtracting rational expressions: like denominators
- Intro to adding & subtracting rational expressions
- Add & subtract rational expressions: like denominators
- Intro to adding rational expressions with unlike denominators
- Adding rational expression: unlike denominators
- Subtracting rational expressions: unlike denominators
- Add & subtract rational expressions (basic)
- Least common multiple
- Least common multiple: repeating factors
- Least common multiple
- Subtracting rational expressions: factored denominators
- Least common multiple of polynomials
- Adding & subtracting rational expressions
- Add & subtract rational expressions: factored denominators
- Subtracting rational expressions
- Add & subtract rational expressions

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# Least common multiple

The least common multiple (LCM) is the smallest number that two or more numbers can divide into evenly. To find the LCM, you can use the prime factorization method or list the multiples of each number. Prime factorization involves breaking down numbers into their prime factors and constructing the smallest number with all the factors. Listing multiples involves finding the smallest shared multiple. Both methods help find the LCM of numbers like 18 and 12. Created by Sal Khan.

## Want to join the conversation?

- Is the LCM (Least Common Multiple) useful in real life? If so, could someone provide some examples?(244 votes)
- I'm assuming that if you're baking something, like a really extravagant cake or something, then figuring out the least common multiple just might work when you're trying to figure out how many cartons of eggs to get to satisfy the ingredients. Also, it might work when trying to figure out if a discount is worth it when you're at the supermarket and comparing any of the "2 for $2" or "5 for $5" types of deals. It's almost10:30at night in this part of the world right now and I'm a little tired so I could be wrong, but those are my best guesses! :)(29 votes)

- is the least common factor the same as the lcm?(38 votes)
- No. LCM stands for Least Common Multiple. A multiple is a number you get when you multiply a number by a whole number (greater than 0). A factor is one of the numbers that multiplies by a whole number to get that number.

example: the multiples of 8 are 8, 16, 24, 32, 40, 48, 56...

the factors of 8 are 1, 2, 4, 8.

The term least common factor doesn't really make sense since the least common factor of any pair of numbers is 1. Not exactly a useful piece of knowledge.(123 votes)

- finding the LCM is too hard for me! Can anyone give me advice for remembering?(18 votes)
- All you have to do is list the multiplies of both of the numbers and look for the common number.

Example:

5 and 6

5 = 5, 10, 15, 20, 25,**30**, 35, 40, 45, 50

6 = 6, 12, 18, 24,**30**, 36, 42, 48, 54, 60

The LMC of 5 and 6 is 30.

Example:

10 and 12

10 = 10, 20, 30, 40, 50,**60**, 70, 80, 90, 100

12 = 12, 24, 36, 48,**60**, 72, 84, 96, 108, 120

The LMC of 10 and 12 is 60.

Example:

3 and 7

3 = 3, 6, 9, 12, 15, 18,**21**, 24, 27, 30

7 = 7, 14,**21**, 28, 35, 42, 49, 56, 63, 70

The LMC of 3 and 7 is 21.(68 votes)

- what if the number you have has no multiples?(13 votes)
- There is no number without multiples and factors. For factors, there will always be 1 and itself.

eg: factor of a number d = 1 and d

If the number is 1, then the factor is 1.

For multiples, take any number and multiply it with your number.

eg: multiple of a number y = y * another number x

Hope this helps!(19 votes)

- Can we use other methods to find LCM?(10 votes)
- Good question!

There is a prime factorization method for finding the LCM of a list of two or more numbers.

Prime-factor each number. Then for each prime factor, use the**greatest**number of times it appears in any prime factorization.

Example: Find the LCM of 40, 48, and 72.

40 = 2*2*2*5

48 = 2*2*2*2*3

72 = 2*2*2*3*3

The prime factor 2 occurs a maximum of four times, the prime factor 3 occurs a maximum of two times, and the prime factor 5 occurs a maximum of one time. No other prime factors appear at all.

So the LCM is 2*2*2*2*3*3*5 = 720.

By the way, there is a similar method of finding GCF (or HCF or GCD or HCD, where G means greatest, H means highest, F means factor, and D means divisor), but we use each prime factor the**least**number of times it appears in any prime factorization. In our example, the GCF would be 2*2*2 = 8.

An interesting property of GCF and LCM is that, for two numbers, the product of the numbers always equals the GCF times the LCM. However, this might not be true for three or more numbers.(20 votes)

- Can the Least common multiple also be the greatest common multiple(10 votes)
- Well yes, in a way. For 0 the only multiple is 0, and nothing else. There's been confusion about 1 having no multiples. But, all the
**whole numbers**are multiples of 1.(3 votes)

- what is the difference between multiple and factor?(9 votes)
- They both involve multiplication. Factors are what you multiply to get a number. Multiples are what we get after multiplying the number by an integer (number, not a fraction). Example: 2 x 2 are factors and 4 is the multiple.(3 votes)

- what is the difference between a multiple and a factor?(4 votes)
- Multiples and factors are both about multiplying, as follows..

A multiple is what you get when you multiply a number by other numbers, like, we get 10, 15, 20, 25, 30 and on and on, which are some MULTIPLES of 5, for example.

(A multiple, is bigger than the number.)

A FACTOR, is what you can multiply together to get a number. Like the factors of 24, can be 1, 2, 3, 4, 6, 8, and 12. These are FACTORS of 24, for example, because you can multiply 2 times 12 to get 24. (The factors of a certain number are smaller than the number.(11 votes)

- Hello, why is it that when he factorizes, he only gets picks 1 number from the 12? And why that specific number?(5 votes)
- You can really only divide 12 by one number at a time. The order that you do it doesn't really matter. In the video, Sal went from 12 to 2 and 6, but you could just as easily go from 12 to 3 and 4. Decompose the 4 into 2 * 2 and you get the same result as he did.(9 votes)

- But can't all the numbers be broken down by 1?(4 votes)
- Yes, but that doesn't actually leave us with different numbers, so we ignore divisibility by 1.(3 votes)

## Video transcript

What is the least common
multiple of 36 and 12? So another way to say this is
LCM, in parentheses, 36 to 12. And this is literally
saying what's the least common
multiple of 36 and 12? Well, this one might
pop out at you, because 36 itself
is a multiple of 12. And 36 is also a multiple of 36. It's 1 times 36. So the smallest number that is
both a multiple of 36 and 12-- because 36 is a multiple
of 12-- is actually 36. There we go. Let's do a couple more of these. That one was too easy. What is the least common
multiple of 18 and 12? And they just state this
with a different notation. The least common
multiple of 18 and 12 is equal to question mark. So let's think about
this a little bit. So there's a couple of ways
you can think about-- so let's just write down our
numbers that we care about. We care about 18,
and we care about 12. So there's two ways that
we could approach this. One is the prime
factorization approach. We can take the prime
factorization of both of these numbers
and then construct the smallest number
whose prime factorization has all of the ingredients
of both of these numbers, and that will be the
least common multiple. So let's do that. 18 is 2 times 9, which is
the same thing as 2 times 3 times 3, or 18 is 2 times 9. 9 is 3 times 3. So we could write 18 is
equal to 2 times 3 times 3. That's its prime factorization. 12 is 2 times 6. 6 is 2 times 3. So 12 is equal to
2 times 2 times 3. Now, the least common
multiple of 18 and 12-- let me write this down-- so
the least common multiple of 18 and 12 is going to have to have
enough prime factors to cover both of these
numbers and no more, because we want the least
common multiple or the smallest common multiple. So let's think about it. Well, it needs to have at
least 1, 2, a 3 and a 3 in order to be divisible by 18. So let's write that down. So we have to have
a 2 times 3 times 3. This makes it divisible by 18. If you multiply this
out, you actually get 18. And now let's look at the 12. So this part right over
here-- let me make it clear. This part right over
here is the part that makes up 18, makes
it divisible by 18. And then let's see. 12, we need two 2's and a 3. Well, we already have one 3,
so our 3 is taken care of. We have one 2, so this
2 is taken care of. But we don't have two 2s's. So we need another 2 here. So, notice, now this number
right over here has a 2 times 2 times 3 in it, or it has a
12 in it, and it has a 2 times 3 times 3, or an 18 in it. So this right over here is
the least common multiple of 18 and 12. If we multiply it out,
so 2 times 2 is 4. 4 times 3 is 12. 12 times 3 is equal to 36. And we are done. Now, the other way
you could've done it is what I would say
just the brute force method of just looking at the
multiples of these numbers. You would say, well, let's see. The multiples of 18
are 18, 36, and I could keep going
higher and higher, 54. And I could keep going. And the multiples of
12 are 12, 24, 36. And immediately I say, well, I
don't have to go any further. I already found a
multiple of both, and this is the smallest
multiple of both. It is 36. You might say, hey,
why would I ever do this one right over here
as opposed to this one? A couple of reasons. This one, you're
kind of-- it's fun, because you're actually
decomposing the number and then building it back up. And also, this is a better
way, especially if you're doing it with really, really
large and hairy numbers. Really, really, really
large and hairy numbers where you keep trying to
find all the multiples, you might have to go pretty
far to actually figure out what their least
common multiple is. Here, you'll be able to do it a
little bit more systematically, and you'll know
what you're doing.