If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Least common multiple

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?

  • leaf blue style avatar for user Cameron Christensen
    Is the LCM (Least Common Multiple) useful in real life? If so, could someone provide some examples?
    (232 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Michael Wais Jr
      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 almost at 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! :)
      (14 votes)
  • hopper jumping style avatar for user Tyler
    is the least common factor the same as the lcm?
    (33 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user Christi
      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.
      (114 votes)
  • starky tree style avatar for user Jae
    finding the LCM is too hard for me! Can anyone give me advice for remembering?
    (15 votes)
    Default Khan Academy avatar avatar for user
    • leaf blue style avatar for user Anthony Jacquez
      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.
      (61 votes)
  • starky sapling style avatar for user Cassandra Martin
    what if the number you have has no multiples?
    (11 votes)
    Default Khan Academy avatar avatar for user
    • hopper cool style avatar for user Noureen Aneeze
      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!
      (14 votes)
  • male robot hal style avatar for user TheOGTristan
    Can the Least common multiple also be the greatest common multiple
    (10 votes)
    Default Khan Academy avatar avatar for user
  • winston baby style avatar for user justvicky :3
    Can we use other methods to find LCM?
    (7 votes)
    Default Khan Academy avatar avatar for user
    • primosaur seed style avatar for user Ian Pulizzotto
      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.
      (11 votes)
  • starky sapling style avatar for user julia
    what is the difference between multiple and factor?
    (8 votes)
    Default Khan Academy avatar avatar for user
  • mr pants teal style avatar for user katrina
    what is the difference between a multiple and a factor?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • mr pants teal style avatar for user Karen Copstead
      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.
      (9 votes)
  • duskpin ultimate style avatar for user Kiba
    Hello, why is it that when he factorizes, he only gets picks 1 number from the 12? And why that specific number?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • primosaur tree style avatar for user George Soilis
    I just realised the "irony" in the name Least common multiple(that was the source of my confusion since i am not a native english speaker)..
    least doesnt mean it is smaller than the numbers offered by the problem..:):)
    (4 votes)
    Default Khan Academy avatar avatar for user

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.