Where math comes alive


Ever need to find the LCM (same as the LCD) for a pair of two numbers, but you don’t feel like spending two hours writing out the multiples for the numbers and waiting till you get a match.

Of course you need to do this — a lot!  Example:  whenever you add fractions with different denominators you need to find the common denominator. That is the LCM.

Here’s a quick way to do this.

The only way to teach this is by example, so that’s what I’ll do — by finding the LCM for 18 and 30.

Step 1)  Find the GCF for the two numbers.

For 18 and 30, GCF is 6.

Step 2)  Divide that GCF into either number; it doesn’t matter which one you choose, so choose the one that’s easier to divide.

Choose 18. Divide 18 by 6. Answer = 3.

Step 3)  Take that answer and multiply it by the other number.

3 x 30  =  90

Step 4)  Celebrate …

… because the answer you just got is the LCM. It’s that easy.

Note:  if you want to check that this technique does work, divide by the other number, and see if you don’t get the same answer.

 

PRACTICE:  Find the LCM (aka LCD) for each pair of numbers.

a)  8 and 12
b)  10 and 15
c)   14 and 20
d)  18 and 24
e)  18 and 27
f)  15 and 25
g)  21 and 28
h)   20 and 26
j)   24 and 30
k)  30 and 45
l)  48 and 60

ANSWERS:

a)  8 and 12; LCM =  24
b)  10 and 15; LCM =  30
c)   14 and 20; LCM =  140
d)  18 and 24; LCM =  72
e)  18 and 27; LCM =  54
f)  15 and 25; LCM =  75
g)  21 and 28; LCM =  84
h)   20 and 26; LCM =  260
j)   24 and 30; LCM =  120
k)  30 and 45; LCM =  90
l)  48 and 60; LCM =  240

Once you learn this trick, have fun using it, as it is a real time-saver!

Josh Rappaport is the author of the Algebra Survival Guide and Workbook, which comprise an award-winning program that makes algebra do-able! The books break algebraic concepts down into manageable chunks and provide instruction through a captivating Q&A format. Josh also is the author of PreAlgebra Blastoff!, which presents an engaging, hands-on approach (plus 16-page color comic book) for learning the rules of integers. Josh’s line of unique, student-centered math-help books is published by Singing Turtle Press and can be found on Amazon.com

Comments on: "How to Find the LCM – FAST!!!" (43)

  1. THNX, you can’t blv how mch time this sves! <3

  2. Hi. Nice trick, will be very helpful for competitions. But what if there are more than two numbers or a bigger number like for example 3190 or any may be any other number.

    Would this trick help in solving such problems?

    Regards

  3. praveen said:

    but how to find it for greater than 2 numbers
    like for
    4,6,8

    6,9,7

    6,9,17,8
    ?

    • Praveen,

      The easiest way to find the LCM for two or more numbers is this.

      First, prime factorize the three numbers.

      Secondly, line up the prime factors.

      Thirdly, for each prime number, choose the largest factor. Call these the “LCM factors.”

      Fourthly, multiply the LCM factors together. The product will be the LCM of the various numbers.

      Example, find the LCM for 12, 15, 24

      Prime Factorizations:

      12 = 2^2 x 3
      15 = 3 x 5
      24 = 2^3 x 3

      LCM factors:
      for 2, it is 2^3 = 8
      for 3, it is 3^1 = 3
      for 5, it is 5^1 = 5

      Product of LCM factors: 8 x 3 x 5 = 120
      So the LCM for the three numbers = 120

      Hope that helps!

      • i cant find the lcm of 11 and 13 i think ill leave that way cuz i couldnt find it

      • Hello,

        The LCM for 11 and 13 is extremely easy to find, so you don’t need any kind of trick for it.
        When looking for the LCM of two numbers, both of which are prime, you find it simply
        by multiplying the two numbers together. So the LCM for 11 and 13 is just 143 (11 x 13) since both
        11 and 13 are prime.

        In fact, the trick I just explained works in a broader sense. Whenever the larger of
        the two numbers is prime, you just multiply the two numbers together to find
        the LCM. So for example, you would just multiply the two numbers to find
        the LCM for:

        a) 12 and 17
        b) 16 and 31
        c) 24 and 37

        since for each pair, the larger number (17, 31, 37) is
        prime.

        Hope that helps!
        —  Josh

      • Hey Josh,

        Thank you SO MUCH for the tips and tricks. I find it a HUGE time saver when solving mathematics by hand. My professor insists I solve using the original method, but I believe anyone would prefer a short cut to a problem that will get them the exact same result. Now if you can just figure out a short cut to become wealthy…

        Cheers,

        Jon

  4. thank you sooooooooo much I like it! you just make my day.

  5. Wow, this is extremely helpful! Thank you! :D I discovered this trick by myself, but I forgot it. :/ Once again, thank you! :D

  6. find 2 numbers so that 30 is the lowest common multiple of the two numbers, neither of the numbers maybe 30.
    i would be really greatful for info :)

  7. Yo, this trick has really helped a lot. Thank-you. Just wanted to ask if you know of any other easy tricks to do conversion problems? Also, as you said ‘if the larger number of the two is a prime number you multiple the two together’ does that ALWAYS work?

    Regards. x

    • To answer your last question first, yes, this is always true. If you’re finding the LCM for any two numbers, and the larger number is prime, you ALWAYS find the LCM by just multiplying the two numbers together. To answer your previous question, here’s another trick. If you’re finding the LCM for two numbers, and those numbers have NO COMMON FACTORS — like 4 and 15; or 9 and 25 — again you get the LCM by just multiplying the two numbers together. The reason is that two numbers have an LCM that’s less than their product ONLY IF THEY HAVE A COMMON FACTOR.

  8. Wow! thanks so much for your tip. I’m having a little trouble following your advice about finding the LCM for more than 3 numbers. I saw it above but I got lost after finding the prime factorizations. Could you help?

  9. Hi Gabriella,

    I don’t know what else I can tell you, other than what I said in my post. You prime factorize all of the numbers. Then for each prime factor, you take the factor with the largest exponent. For example, if you have factors of X^3, x^5 and x^4, you would use the x^5 factor, since 5 is the largest of the three exponents: 3, 4 and 5. Then, once you have those factors, you just multiply them together to get the LCM. If you are still puzzled by this, perhaps it would help if you share a specific problem for which you are unsure how to find the LCM.

  10. This stuff is really awesome
    at first i wad just looking for a site that would tell me the answer but its so cool how u actualy explaimed everything josh.
    so i just wanted to say good job

  11. itz amazing

  12. thanx

  13. Yanna Bannana said:

    COOL!:D thanks i got perfecto in our long test, great work!!

  14. [...] The information in this video dovetails with the info in this post. [...]

  15. what is the lcm of 72 and 120???

  16. Wow that was strange. I just wrote an extremely long comment but after I
    clicked submit my comment didn’t show up. Grrrr… well I’m not writing all that over again.
    Anyway, just wanted to say fantastic blog!

  17. Rainbow Dash ⚡ said:

    Thanx a lot! Your tips really helped! 😀

  18. Awesome! You just helped me work this out, now I can sleep!

  19. Yes, sign up is free for both the blog and my newsletter. You can sign up for the blog on this page, and sign up for the newsletter at: http://bit.ly/Zm3Cuv Thanks for asking.

  20. Vishesh said:

    Hey josh
    Great site. Thanks a lot. Solved a lot of problems.
    I have one request. Can you show easier way to find GCF too.
    Thanks

  21. It’s not my first time to pay a quick visit this web site, i am visiting this web site dailly and get pleasant data from here everyday.

  22. thank you very much you help me in my math test

  23. Can you please solve this 48 and 60; LCM = 240? because my answer was 480, so do we always need to simplify the answer?

    • Hi Simara,
      The correct answer is 240. With my technique, you use the GCF, which is 12. 12 goes into 48 4 times. Take that answer, 4, and multiply it by the other number, 60. 4 x 60 = 240, and that is the LCM. You can check it and see. 48 goes into 240 5 times. And 60 goes into 240 4 times. And there’s no number smaller than 240 that both 48 and 60 go into. So that’s it.

  24. Kylie Grambart said:

    what is the LCM of 25 and 37???

  25. Regina Cheung said:

    um… how do you do it with 3 numbers? Do you just like divide 2 numbers by the gcf and times it all together by the 3rd number? Or do you divide once and multiply it by both numbers and add the products together?

    • Hi Regina,

      I have put up a blogpost on that topic. Go to the Blogroll (right side of the main blog page) and scroll down till you see: LCM with 3+ Numbers.

      That takes you to the post that shows how to find the LCM for three or more numbers. I hope you find it helpful!

      I also made a YouTube video on this topic, and you will find it here:

  26. Regina Cheung said:

    that does help, but… i know that way already, but was just wondering if there was an easier way.

    • Hi Regina,

      Here’s something that should help. Suppose we want the GCF for 18, 45 and 54. Here’s what you do.

      Look at the differences between the three numbers (one pair at a time) and find the smallest difference. For these numbers, the differences are: 27, 36, 9. Thus 9 is the smallest difference. Interestingly (and helpfully), this tells you that 9 is the largest number that could possibly be the GCF. I have dubbed this the GPGCF (Greatest Possible GCF). But it gets even better. For, as it turns out, the only numbers that could possibly divide evenly into all three numbers are the FACTORS of the GPGCF. So for this example, you simply test the factors of 9 from largest to smallest. For 9, you would obviously test only 9 and 3. But since 9 does go into 18, 45 and 54, you can stop there, assured that 9 is the GCF. And in general, as long as you test the factors of the GPGCF in order from largest to smallest, the first GPGCF factor that divides into all three numbers must be the GCF. Pretty cool, huh?

      There is also a standard, textbook way to find the GCF for a set of three or more numbers, and I’m guessing you know it. But in case you don’t, this technique involves prime factorizing all of the numbers, then taking the largest common prime factors and multiplying them together.

      Let’s do this for the same numbers: 18, 45 and 54. When you prime factorize these, you get: 18 = 2 x 3^2; 45 = 5 x 3^2; and 54 = 2 x 3^3. From this you can see that 18, 45 and 54 do not all share a power of 2 or a power of 5. However they do share a power of 3. One number has a 3^3, but they don’t all have a 3^3. However they do all have 3^2. Therefore 3^2 = 9, is the GCF for these three numbers.

      Hope this helps!

      Kind regards,
      — Josh

      • Regina Cheung said:

        I understand how you got 2 differences for the first way: 54-45 and 45-18. So how did you get a third difference?

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