Kiss those Math Headaches GOODBYE!


Is there a quick-and-easy way to find the LCM for three or more numbers … WITHOUT prime factorizing?

Of course! We’ll demonstrate the technique by finding the LCM for 10, 14, 20.

High-Octane Boost for Math

High-Octane Boost for Math Ed

To begin, use the technique for finding the GCF for 10, 14, 20 that’s shown in my post:  How to Find GCF for 3+ Numbers — FAST … no prime factorizing. If you don’t want to go to that post, no worries. I’ll re-show the technique here.

1st)   Write the numbers from left to right:

……….   10     14     20

[The periods: …… are just to indent the lines. They have no mathematical meaning.]

2nd)  If possible, rip out a factor common to all numbers. The factor 2 is common. So divide the three numbers by 2 [10 ÷ 2 = 5 and 20 ÷ 2 = 10] and show the result below:

2   |       10     14     20
……….   5      7      10

3rd)  At this point, notice there’s no number that goes into the remaining numbers: 5, 7, 10. That means you’ve found that the GCF is the number pulled out, 2. At this point we’re at a crossroads. We’re done finding the GCF, but now we’re at the start of a new process, finding the LCM.
To proceed toward getting the LCM, see if there’s any number that goes into any pair of remaining numbers. Well, 5 goes into 5 and 10. So divide both those numbers by 5 [5 ÷ 5 = 1 and 10 ÷ 5 = 2] , and show the results below:

2   |       10     14     20
5   |         5      7      10
………..  1       7       2

Notice that if there’s a number 5 doesn’t go into, you leave that number as is. So leave the 7 as 7.

4th)  Repeat. See if there’s a number that goes into two of the remaining numbers. Since nothing goes into 1, 7, and 2, we’re done. To get the LCM, multiply all of the outer numbers. That means you multiply the numbers you pulled out on the left (2 and 5), and also multiply the numbers at the bottom (1, 7 and 2). Ignoring the meaningless 1, you have:  2 x 5 x 7 x 2 = 140, and that’s the LCM.

To see the process in more depth, let’s find the LCM for … not three, not four … but five numbers:
6, 12, 18, 30, 36.

1st)   Write the numbers left to right:

………  6     12     18     30     36

2nd)  If possible, rip out a common factor.  2 is common, so divide all by 2 and show the results below:

2     |    6     12     18     30     36
………. 3      6       9     15     18

3rd)  Repeat. See if there’s a number that goes into the five remaining numbers. 3 goes into all, so divide all by 3 and show the results below:

2     |    6     12     18     30     36
3     |    3       6       9     15     18
……..   1       2       3       5       6

4th)  Repeat. See if any number goes into the last remaining numbers. Nothing goes into all of them, so now you get the GCF by multiplying the left-hand column numbers. GCF = 2 x 3 = 6.
Proceeding to find the LCM, look for any number that goes into two or more of the remaining numbers. One such number is 3, which goes into the remaining 3 and 6. Divide those numbers by 3 and leave the other numbers as they are.

2     |    6     12     18     30     36
3     |    3       6       9     15     18
3     |    1       2       3       5       6
……… 1       2       1       5       2

5th)  Interesting! Notice that 2 goes into the two remaining 2s, so pull out a 2 and show the results below:

2     |    6     12     18     30     36
3     |    3       6       9     15     18
3     |    1       2       3       5       6
2     |    1       2       1       5       2
……..   1      1        1       5       1

6th)  We’ve whittled the bottom row’s numbers so far down that finally there’s no number that goes into two or more of them (except 1, which doesn’t help). So we have all the numbers we need to find the LCM. Multiply them together. The left column gives us:  2 x 3 x 3 x 2. The bottom row gives us 1 x 1 x 1 x 5 x 1. Multiply all of those (non-1) numbers together, you get:
2 x 2 x 3 x 3 x 5 = 180, and that is the LCM! Pretty amazing, huh? And no prime factorizing, to boot.

Some people find that this process takes a bit of practice to get used to it. So here are a few problems to help you become an LCM-finding expert!

a)  12, 18, 30
b)   8, 18, 24
c)  15, 20, 30, 35
d)  16, 24, 40, 56
e)   16, 48, 64, 80, 112

And the answers. LCM for each set is:

a)   180
b)   72
c)   420
d)   1680
e)   6720

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Comments on: "How to Find the LCM for 3+ Numbers — FAST!" (12)

  1. This is great. I always joke that I wish the Euclidean Algorithm would be taught in schools, but I like this even better.

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  2. This is amazing. If this method were taught in schools when GCFs and LCMs are introduced, I have a feeling a lot more students would struggle a lot less with factoring and fractions.

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  3. Thank you so much…..you can’t imagine the extent to which this method helped me, I truly learned something…. I’m forever in debt to you

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  4. Thank you for your help!

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  5. This is fantastic! What a great way to teach students. Thank you so much for sharing!

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  6. Thanks! This was a BIG help
    @-Isha

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  7. What if your numbers have NO common factor??? :O

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    • Then the LCM is simply the product of the numbers. Example. if the numbers are 11, 13 and 17, the LCM is the product of 11, 13 and 17, which is 2,431. Moral of the story: the more common factors the numbers have, and the greater those common factors are, the smaller their LCM. And vice-versa, the fewer common factors they have, and the smaller those common factors are, the greater their LCM.

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  8. Thank you very much! Very helpful!

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  9. Brintha shivakumar said:

    Atlast I found the easy way to teach my little one. Very easy to understand. Thank you.

    Like

  10. Your Niece said:

    This way really worked better than my pre-algebra book!!

    Like

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