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Math Basics — Least Common Multiple

If GRE math concepts were people, Least Common Multiple (LCM) would hardly be the most noticeable person in the room. After all, we have the really big personalities who always elbow for attention: Probability, Work Rates and Combinations (none of whom would win the popularity context). Then, there are Exponents standing together menacingly with Coordinate Geometry, with Scientific Notation sneering up at us. Finally, squished in the corner, making room for all these other concepts, is our little wallflower: Least Common Multiple. So self-effacing—introducing itself with a Least—and meek, this concept is more familiar in its diminutive: LCM.

But let’s not forget that finding the Least Common Multiple can help us on questions ranging from work rates to number properties. So, now is our cue to walk up to meek LCM and say hello.

First off, finding LCM for a group of numbers requires three steps.


1. Find the Factors that the Numbers Have in Common.

Take a look at the following numbers: 14, 35 and 70.

Which factors do they have in common? To answer this question, we need to break down each number into its prime factors.

14 = 7 x 2

35 = 5 x 7

70 = 5 x 7 x 2

All three of them have the number 7. Also, 35 and 70 have 5 in common, and 14 and 70 can both be divided by 2. Therefore, the shared factors are 7, 5, and 2. Note that we include each shared factor only once, even if, as is the case for the number 7, it shows up in all three numbers.

2. Find the Lone Factors.

Take a look at the following numbers: 15, 8, and 12.

Let’s break down each to prime factors:

15 = 3 x 5

8 = 2 x 2 x 2

12 = 2 x 2 x 3

Is there any factor that is not shared by the other two factors? Most saliently, there is the one 5. Are there any other lone factors? Actually, there is one 2, all by itself. Therefore, the lone factors are 5 and 2. Notice that two of the 2s overlap for both 8 and 12. But the number 8 has three factors of 2. Therefore, that extra two is a lone factor.

Let’s now find the shared factors (step 1) of 15, 8, and 12, so we can go on to the third step. We can see that the shared factors are 2, 2, and 3.

3. Multiply the Product of the Shared Factors by the Product of the Lone Factors

Shared Factors : 2 x 2 x 3 = 12

Lone Factors: 2 x 5 = 10

12 x 10 = 120.

Therefore the LCM is 120.

And just like that we’ve met, not the life of the party, but an indispensable concept to doing well on the GRE math section.

By the way, students who use Magoosh GRE improve their scores by an average of 8 points on the new scale (150 points on the old scale.) Click here to learn more.

13 Responses to Math Basics — Least Common Multiple

  1. Gaurav Verma June 12, 2016 at 1:14 pm #

    Hi..! I had this query. It says we shouldn’t multiply numerators before cancelling if possble in the formula: LCM=P*Q/GCF. Is there any good reason for the same??

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert June 14, 2016 at 7:21 am #

      Hi Gaurav,

      Good question! We want to cancel first to simplify the overall calculation. When we cancel factors in the numerator and denominator, we’re left with smaller numbers to multiply together, making finding the product in the numerator easier. Let’s look at a very clear example to see what I mean 🙂 Say P = 12 and Q = 48. The GCF for these two numbers is 12. So,

      LCM = 12*48/12

      If we were to multiply the numerator first, we would have a more complicated product to find:

      12*48 = 576

      Then, we would need to divide this large number by 12 to find the LCM:

      576/12 = 48

      On the other hand, if we cancel first, we simplify the numerator significantly, as we can divide 12/12 = 1:

      12*48/12 = 1*48/1 = 48

      In fact, we do not need to carry out any other calculations in order to determine that the LCM of 12 and 48 is 48.

      Now, it’s not always the case that we can cancel out one of the terms completely in the numerator. However, the example illustrates why canceling before multiplying out the numerator is good to do.

      Hope this helps 🙂

  2. Mary Ann July 29, 2015 at 1:53 pm #

    In McGraw-Hill’s “Conquering the New GRE Math” finding the LCM is taught differently than on the Magoosh videos. It seems simpler. They say to first find the prime factorization of each number. “Then you need to find the greatest power each different factor of the number has. The least common multiple will be the product of each different factor to the greatest power it occurs.” (p.56)

    For example: Find the LCM of 150 and 225
    The different prime factors are 2,3, and 5. The greatest power of 2 is 1, the greatest power of 3 is 2, and the greatest power of 5 is 2.
    LCM (150,225)= 2^1*3^2*5^2= 450

    Is there a reason it is not taught on Magoosh this way?

    • Chris Lele
      Chris Lele July 31, 2015 at 1:24 pm #

      Hi Mary Ann,

      I don’t think there really is any good reason why Magoosh taught prime factorization this way. I agree it is a little more complicated. The McGraw Hill way is slightly more straightforward. That said, it is another way of conceiving of the problem and so it might help give a deeper understand of LCM.

      But if the McGraw Hill works for you, definitely stick with it 🙂

      • Mary Ann August 1, 2015 at 10:06 am #

        Thanks! That is actually really helpful.

        I’ve been very impressed with the Magoosh videos. The videos have made learning math again actually enjoyable instead of overwhelming. Much thanks to you and your team!

        • D January 12, 2016 at 11:24 pm #

          Khan Academy also explains calculating LCM in the same way that Mary Ann described; and I find that method faster to calculate. Still, I enjoyed Chris’s explanation as it gives a different perspective for LCMs!

  3. Meera January 18, 2015 at 11:00 am #

    What an intro to LCM! I have always thought math was a chore but I must say practicing with magoosh has made me look at math differently and I’m thoroughly enjoying it. Hope I get a good score 🙂

  4. Valeria April 15, 2013 at 8:35 pm #

    Hi. My problem with LCM and GCF is that I don’t know when to use each. For example, in this problem: “What is the smallest number that is divisible by both 35 and 36”, how can I tell if they are asking me to calculate the LCM or the GCF? Thanks.

    • Chris Lele
      Chris Lele April 16, 2013 at 4:15 pm #

      Great question!

      With LCM always think “bigger”, as in the number is going to be bigger than the numbers you are working with. This is slightly counterintuitive because the L stands for least. As long as you remember that LCM stands for “bigger” you are fine.

      In this case, we are looking for a number that is divisible by two smaller numbers. Therefore, we are looking for a bigger number. No we use LCM.

      GCF, on the other hand, looks for the greatest factor that some numbers have in common. A factor can never be greater than the least number of the set. For instance, if I have 24, 42, 54, the greatest factor these numbers share can’t be greater than 24. (It happens to be 6).

      Hope that helps :)!

      • Valeria April 16, 2013 at 6:42 pm #

        Your explanation helped me a lot. Thank you very much.

        • Chris Lele
          Chris Lele April 17, 2013 at 12:46 pm #

          You are welcome :)!

  5. mahmoud November 15, 2011 at 12:52 am #

    Thank you very much, you helped me very much with GRE math.

    • Chris Lele
      Chris November 15, 2011 at 3:29 pm #

      Great Mahmoud,

      I’m happy this was helpful!

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