How to find the LCM — Intuitive Method

Recently I’ve been interested in discovering a cool, new way to get the LCM for a pair of numbers.

Criteria:  Method that is short and sweet. Even more important, a method that makes sense INTUITIVELY.

I can’t speak for any of you , but I’ve always felt that the standard techniques for finding the LCM DON’T make good intuitive sense. So I was trying to find a procedure that could harmonize with mathematical “common sense.”

So, after spending time searching,” I’m happy to report that I’ve found what I was looking for.

Today’s post is all about this new way of finding the least common multiple. A way that is, for me and hopefully you too,  more intuitive!

This method does take some time and space to explain — as you can see by the length of this post. But don’t let that intimidate you. Once you understand this new method, you’ll find that it is actually fairly quick and easy, as you will surely see by looking at the brevity of Examples A, B, and C at the end of this post.

To describe the method, I first need to define a few terms and explain a few ideas. So please bear with me. I’ll do that now, using, as an example, the numbers of 8 and 12, as if we were trying to find the LCM for that pair of numbers.

DEFINITIONS:  We’re finding the LCM for a pair of numbers. To make it clear which of the two number we’re referring to at any given time, we’ll either say the “smaller number” or the “larger number.” In our example, 8 would be the smaller number, 12, the larger number.

One thing that the new method looks at is the difference between the two numbers. For 12 and 8, that difference is 4, for the simple reason that 12 – 8 = 4. For any pair of numbers, I’ll be calling this difference the  ”gap.”

The new method also takes a look at the first multiples (M’s, in my shorthand) of the two numbers.  By multiples, I mean what we get when we multiply both of the original numbers by any of the counting numbers:  1, or 2, 3, etc.  So …

1st M’s means “first multiples” — 8 & 12 in our example.

2nd M’s means “second multiples” — 16 and 24 in our example,
since 8 x 2 = 16, and 12 x 2 = 24

3rd M’s means “third multiples” — 24 and 36 in our example,
since 8 x 3 = 24, and 12 x 3 = 36

With all of this, we produce the following chart for 8 and 12:

1st M’s          2nd M’s          3rd M’s          4th M’s          5th M’s

8                      16                      24                    32                 40

12                    24                      36                    48                 60

One last point. The new method deals with the difference between the numbers for any pair of multiples. So for each vertically aligned pair, we  look at what you get when you subtract the smaller number from the larger. Example for the 4th M’s:  larger number = 48; smaller number = 32. Difference is 16, since  48 – 32 = 16.

To show this difference for each and every pair of multiples, we add a bottom row. With the bottom row displaying the difference for each pair, here is the completed chart:

1st M’s          2nd M’s          3rd M’s          4th M’s          5th M’s

8                      16                      24                    32                 40

12                    24                      36                    48                 60

4                      8                       12                     16                 20

[Note to "wiseguys" in the crowd. In my mind's ear, I hear a class clown saying: "But if you make this whole chart, you're going to SEE the LCM right at the beginning. So therefore you don't need the 'method.' " As you'll see, though, after you learn this technique, you do NOT produce this chart. The purpose of the chart is just to SHOW that the process makes sense. "Wiseguys" are great, though, don't you think?]

Back to the chart, you might be wondering why we’re scrutinizing the gap and the differences? What could any of that have to do with the LCM? Good question. It turns out that the gap and differences actually show us when we reach the LCM. Here’s how …

Since the smaller number is 8, it’s obvious that every new multiple of this number has to be 8 greater than the previous multiple. And we can see this in the chart as we look at the row beginning with 8. Each term is 8 more than the previous term: 8, then 16, then 24, then 32 — etc. Now think about the differences. If we can find the first column where the difference is also 8, we will have found the LCM. It takes a second of reasoning to grasp this. If the difference is 8, that means the smaller number is 8 less than the larger number in that vertical pair (the number below). But as we just saw, in the next column to the right, the smaller number will be exactly 8 more than it is now (the number to the right). In other words, where the difference is 8, the next value of the smaller number will be the current value of the larger number.

How, then, can we see this in the chart? Well, just look for the M’s that have a difference of 8? Not too hard … the difference is 8 at the 2nd M’s. where the smaller number is 16, and the larger number is 24. From what we just said, the next value of the smaller number must be the same as thecurrent value of the larger number. And indeed it is:  the next value of the smaller number is 24 at 3rd M’s. And the current value of the larger number is also 24 at 2nd M’s. So 24 has to be the LCM.

In a way this is so simple, so obvious, that it’s almost hard to spot  (like trying to look out and see your own nose — without a mirror!).

To make it easier to see the idea, let’s look at another quick example. Suppose this time that we’re seeking the LCM for 12 and 15. So now the smaller number = 12; larger number = 15; gap = 3. Chart looks like this:

1st M’s          2nd M’s          3rd M’s          4th M’s          5th M’s

12                      24                      36                    48                60

15                     30                      45                    60                 75

3                       6                          9                     12                  15

We’re looking for a difference that is equal to the smaller number, 12. Where does this happen? At the 4th M’s, for here the difference = 12. That means that when we get to the smaller number’s next multiple at the 5th M’s, which is 60, it will have to be the same as the 4th M’s value of the larger number. And indeed it is. The value of both of these numbers = 60, so 60 is the LCM for 12 and 15.

This is a bit of a tongue-twister, perhaps, but it is actually conceptually pretty simple. Right?

You might be wondering, though, how we can predict when the difference will be the same as the smaller number? If we could predict this, we could find the LCM with ease!

Turns out there is an easy way to predict when this happens. Making this prediction relates to the gap and the way that the gap makes the difference gradually but steadily increase as we read the chart left to right.

Look at the bottom row to see how the differences increase left to right:  3 — 6 — 9 — 12 — 15 …

These values go up by 3 with each new pair of multiples. And small wonder … with a gap of 3, the differences increase by 3 for each new pair of multiples.

So the differences start at 3 and go up by 3 with each new multiple. How, then, do we figure out when that difference will be 12? This is like a 4th grade word problem. We just divide 12 by 3, and we get 4. This means that for this problem, the difference will be 12 at the 4th Ms. Then we look at the larger number’s value at the 4th M’s, and we see that it’s 60. That tells us that the LCM is 60.

To summarize:  the LCM is the value of the larger number at the multiple where the difference IS the smaller number. Plain and simple.

To formalize all of this, here’s what you need to do — easy as 1-2-3:

1st)  Find the gap.

2nd)  Divide the smaller number by the gap, and notice the quotient.

3rd)  Multiply the quotient by the larger number. The answer you get is the LCM.

Let’s try a few more so you can get the hang of this.

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EXAMPLE A —

Problem:  Find LCM for 16 and 20.

1st)  Gap = 4

2nd)  (Sm #) ÷ (Gap)  =  16 ÷ 4 = 4, so Quotient = 4

3rd)  (Quotient) x (Lg #) = 4 x 20 = 80 = LCM

Answer:  LCM for 16 and 20 is 80.

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EXAMPLE B —

Problem:  Find LCM for 21 and 28.

1st)  Gap = 7

2nd)  (Sm #) ÷ (Gap)  =  21 ÷ 7 = 3, so Quotient = 3

3rd)  (Quotient) x (Lg #) = 3 x 28 = 84 = LCM.

Answer:  LCM for 21 and 28 is 84.

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EXAMPLE C —

Problem:  Find LCM for 60 and 72.

1st)  Gap = 12

2nd)  (Sm #) ÷ (Gap)  =  60 ÷ 12 = 5, so Quotient = 5

3rd)  (Quotient) x (Lg #) = 5 x 72 = 360 = LCM.

Answer:  LCM for 60 and 72 is 360.

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I should mention before signing off today that there is one type of situation that I have not yet accounted for. In other words, there’s one kind of situation where we have to do one additional step to find the LCM. Fortunately, though, it is not terribly complicated. And I’ll explain that situation — and tell you how to handle it — in my next post on this topic.

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