Find the LCM (aka LCD) in Two Easy Steps

This is really the “Week of the LCM” for me.

Just as I was finishing my last post, on a new way to find the LCM for a pair of numbers, I discovered another way to do the same thing.

I was looking at the problems at the end of my last post, these problems:

b) 15 and 20; LCM = 60

c) 18 and 20; LCM = 180

d) 24 and 28; LCM = 168, ….

… when I noticed something.

If, in the first problem, you take 15 and 20 and make it into a fraction, you would get 15/20, which reduces to 3/4.

Then if you take the 3 in the numerator of 3/4 and multiply it by 20, you get 60, the LCM.

Similarly, if you take the 4 in the denominator of 3/4 and multiply it by 15, you also get 60.

I thought about this a bit and realized that it leads to another way to find the LCM for any two numbers.

The steps work as follows:

1st) Write the two numbers as a fraction, with the smaller number as numerator. Then reduce the fraction to lowest terms.

2nd) Multiply the original fraction by the reciprocal of the reduced fraction. The fraction that you wind up with has the LCM as both the numerator and denominator.

See how the two steps work with the next problem: Numbers are 18 and 20.

1st) 18/20 = 9/10 (reciprocal = 10/9)

2nd) 18/20 x 10/9 = 180/180, so 180 is the LCM.

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One more example, with the somewhat larger numbers of 24 and 44.

1st) 24/44 = 6/11 (reciprocal = 11/6)

2nd) 24/44 x 11/6 = 264/264, so 264 is the LCM.

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It is that easy. If you’d like to “get a handle” on this project, try these practice problems using this technique.

PROBLEMS:

a) 6 and 8

b) 4 and 10

c) 9 and 15

d) 10 and 16

e) 14 and 21

f) 18 and 45

g) 24 and 28

h) 27 and 63

i) 32 and 48

j) 45 and 55

ANSWERS:

a) 6 and 8; RECIPROCAL = 4/3, LCM = 24

b) 4 and 10; RECIPROCAL = 5/2, LCM = 20

c) 9 and 15; RECIPROCAL = 5/3, LCM = 45

d) 10 and 16; RECIPROCAL = 8/5, LCM = 80

e) 14 and 21; RECIPROCAL = 3/2, LCM = 42

f) 18 and 45; RECIPROCAL = 5/2, LCM = 90

g) 24 and 28; RECIPROCAL = 7/6, LCM = 154

h) 27 and 63; RECIPROCAL = 7/3, LCM = 189

i) 32 and 48; RECIPROCAL = 3/2, LCM = 96

j) 45 and 55; RECIPROCAL = 11/9, LCM = 495

Josh Rappaport is the author of the award-winning Algebra Survival Guide and several other supplemental math books. To check out these products, follow the links in the sidebar, or just visit the Singing Turtle Press website.

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One Response to “Find the LCM (aka LCD) in Two Easy Steps”

  1. Shana Donohue says:

    This post and method are awesome. WAAAY easier for kids, especially those who don’t like Math so much, to not spend a million years finding the LCD. Sweeeet!

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