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.**

– – – – – – – – – – – – – – – – – – –

**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.**

– – – – – – – – – – – – – – – – – – –

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 = 168

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 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 (aka LCD) in Two Easy Steps" (14)Whit Fordsaid: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.

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Josh Rappaportsaid: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.

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Algebra Intro 12: Adding and Subtracting Fractions | Learning and Teaching Mathsaid:[…] slight shortcut to the above approach was described here by fellow blogger Josh Rappaport. If you create a fraction using your two denominators, then […]

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Gopal rawatsaid:it works very well with finding lcm of 2 nos……..but does it work for more than two nos?????

if yes then plz explain how?????

by d way thnx for this shortcut method…..its helping me…..

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Josh Rappaportsaid:Hello Gopal,

I have answered your question in my most recent blog, Find the LCM for Three Numbers.

Hope you find that helpful.

— Josh

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Veenasaid:amazing!!!! And Thankyou!!!

Veena

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azharsaid:did u have a shortcut method of finding hcf also

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Josh Rappaportsaid:Hello Azhar, Yes, I actually have a number of strategies for the hcf, which we more commonly call the gcf in the United States. Here are links to two of my blog posts:

Finding the GCF for two #s, with a video.

Finding the GCF for 3 or more numbers, with a video.

A bit of info on my concept of the GPGCF, the greatest possible greatest common factor, with a video.

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Regina Marysaid:Could you please provide shortcut for finding lcm of 3 numbers?..

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Josh Rappaportsaid:Here is the link for my blog on the shortcut for finding the LCM for three or more numbers.

Hope you find it helpful!

— Josh

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Ashishsaid:Amazing method for 2 numbers..but what if, there were more then two numbers

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Josh Rappaportsaid:Here’s how to find the LCM for three or more numbers.

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Arthisaid:What to do if both the numbers does not have a common factor?

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Josh Rappaportsaid:Great question! If the numbers have no common factor, the task is even easier. The LCM in that case is simply the product of the numbers. Example: 8 and 15 have no common factors since the only factors of 8 are power of 2, and the only factors of 15 are 3 and 5. Therefore the LCM of 8 and 15 is 120. End of story.

Corollary: for any 2 or more numbers that are prime, the LCM is the product of the numbers. Example. Q: What’s the LCM of 7 and 13? A: Since 7 and 13 are both prime, their LCM is their product: 91.

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