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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.

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  • starky tree style avatar for user Jae
    finding the LCM is too hard for me! Can anyone give me advice for remembering?
    (16 votes)
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    • 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.
      (2 votes)
  • male robot hal style avatar for user TheOGTristan
    Can the Least common multiple also be the greatest common multiple
    (11 votes)
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  • winston baby style avatar for user justvicky :3
    Can we use other methods to find LCM?
    (8 votes)
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    • 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.
      (2 votes)
  • 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?
    (1 vote)
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    • piceratops ultimate style avatar for user Nigel Piere
      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)
  • leaf orange style avatar for user hyunjinsong5
    How does this prime factorization method actually work?
    (1 vote)
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    • winston default style avatar for user Dorito
      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)
  • hopper cool style avatar for user Monish Sarkar
    But can't all the numbers be broken down by 1?
    (2 votes)
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  • blobby green style avatar for user Nekisha Rhodes
    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)
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  • blobby green style avatar for user abdullah552002
    what is the LCM of 5, 10, and 15
    (2 votes)
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    • leaf blue style avatar for user Ben Mann
      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)
  • duskpin sapling style avatar for user ☣Correlium☢
    Why does he have to say 2 like 18 times?
    (2 votes)
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  • duskpin ultimate style avatar for user Andrea🦔🤗
    Do you do the same thing to find the GCM?
    (2 votes)
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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.