MathChat

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.

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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