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Least common multiple
CCSS.Math:
Sal finds the LCM (least common multiple) of 12 and 36, and of 12 and 18. He shows how to do that using the prime factorization method, which is a just great! Created by Sal Khan.
Want to join the conversation?
- finding the LCM is too hard for me! Can anyone give me advice for remembering?(16 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.(2 votes)
- Can the Least common multiple also be the greatest common multiple(11 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)
- Can we use other methods to find LCM?(8 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.(2 votes)
- Is the LCM (Least Common Multiple) useful in real life? If so, could someone provide some examples?(1 vote)
- Yes say for example tiles in houses hotels apartments etc. and you want to use the least possible amount of tiles on the floor (Area and perimeter.) So you place a tile on the floor and take a guess of needing 50 tiles (50) (10) least common multiple is 50!(3 votes)
- How does this prime factorization method actually work?(1 vote)
- Prime factorization is basically finding what prime numbers can be multiplied to get whatever number you’re looking at. For example, the prime factorization of 8 is 2*2*2, and if you multiply that, you’re going to get 8. The process is like this:
8 = 4*2
8 = 2*2*2 (the two times two in the beginning makes 4 which can be seen above)(5 votes)
- But can't all the numbers be broken down by 1?(2 votes)
- Yes, but that doesn't actually leave us with different numbers, so we ignore divisibility by 1.(2 votes)
- I still don't understand this. Please if you can, explain it another way? I have failed the mathematics test over eight time☹️(2 votes)
- what is the LCM of 5, 10, and 15(2 votes)
- The LCM of 5, 10 and 15 is 30 as:
Multiples of 5 are 5, 10, 15, 20, 25, 30...
Multiples of 10 are 10, 20, 30, 40, 50...
Multiples of 15 are 15, 30, 45, 60, 75...
Another way of looking at it is 5 = 5*1, 10 = 5*2 and 15 = 5*3, and therefore the LCM will be the smallest number to contain all of these factors which is 5*3*2(*1) = 30.(1 vote)
- Why does he have to say 2 like 18 times?2:52(2 votes)
- Do you do the same thing to find the GCM?(2 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.