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.**

I had never thought of that – great insight!

By forming a fraction from the two numbers and simplifying it, you are getting rid of all the common factors in “one step”, and leaving behind only those factors needed to arrive at a least common multiple. Very elegant and simple step – however I would want to get my students to explain to me why it works before I let them use it.

Thanks for the comment, Whit. I agree that it would be best to either explain why this works, or ask the students if they can figure out why. I am glad that you like this alternative approach for finding the LCM for two numbers.