## Kiss those Math Headaches GOODBYE!

### How to Understand the LCM (Part 1)

I don’t know about you folks, but I’ve always been a bit disappointed by the various techniques for finding the Least Common Multiple (LCM) for a pair of numbers.

While there are several techniques that “work” — by which I mean techniques we can teach to students and have them learn quickly — I’ve known of no technique that makes good intuitive sense. In other words, I’ve known no technique whose underlying principle felt obvious.

Feeling frustrated, I started looking for a technique that would have that undeniable “ring of truth.”

Coffee, Pi and More

And so, after playing around in my “sandbox of numbers” for quite a while,  I’m happy to report that I’ve finally found what I had been looking for.

In today’s post I will show you a way to find the least common multiple that makes sense, at least to me. I hope it will make sense to you as well.

This method takes a bit of time and space to explain — as you can see by the length of this post. But once you understand the method, you’ll find it fairly quick and easy, as you’ll see by looking at the brevity of Examples A, B, and C at the bottom of this post. And to test your understanding, you’ll get a chance to practice this technique at the end, too.

To describe the method, though, I do need to define a few terms and explain a few ideas. I’ll do that now, using, as an example, the numbers 12 and 15. To follow along, suppose that we’re trying to find the LCM for these numbers: 12 and 15.

DEFINITIONS:  We’re finding the LCM for a pair of numbers. To make it clear which of the two numbers we’re referring to at any given moment, we’ll either refer to the smaller number or to the larger number. In our example, 12 is the smaller number, 15 the larger.

One thing that this new method pays attention to is the difference between the two original numbers. For 12 and 15, the difference is 3, for the simple reason that 15 – 12 = 3. For any pair of numbers, we call this difference between the initial numbers the gap.

The new method also takes a look at the first multiples (Ms, in my shorthand) of the two numbers.  By multiples, I mean what we get when we multiply both of the original numbers by the early Natural Numbers of 2, 3, 4, etc.  So …

2nd Ms means second multiples24 & 30 in our example, since 12 x 2 = 24; 15 x 2 = 30.

3rd Ms means third multiples36 and 45 in this example.

4th Ms means fourth multiples 48 and 60, and so on.

We write these labels —  2nd Ms, 3rd Ms, 4th Ms, etc — as column headers in the chart below.

To distinguish the rows in the chart, Ms/S stands for multiples of the smaller number; Ms/L stands for multiples of the larger number.

Another thing that this method pays attention to is the difference between the numbers in each column. For example, the second multiples of 12 and 15 are 24 and 30. So in the 2nd Ms column, the difference is 6, since 30 – 24 = 6. We write these differences in the chart’s bottom row, with Ds being my shorthand for differences. Note that we write the gap in parentheses right after Ds.

With all of this in mind, here’s our chart for 12 and 15:

You might be wondering why we’re paying attention to the differences. It turns out that the differences help us see when we find common multiples in general — and when we find the all-important LCM in particular. Here’s how this works …

As noted, 12 and 15 start out with a gap of 3.

Moving to the 2nd Ms column, the difference becomes 6. In the 3rd Ms column, the difference grows to 9. See the pattern? Because the gap is 3, the difference grows by 3 as we move each additional column to the right.

Then, when we get to the 4th Ms column, something interesting happens. First, notice that the difference in this column is 12, which just happens to equal the smaller number. Looking up in the chart, notice that the multiple of 12 in 4th Ms is 48. Then notice that, since the difference between the smaller and larger number here is 12, the number below 48 is 12 more than 48, which it is, as it is 6o. But then also notice that when you look just to the right of 48, the next multiple is also 12 more than 48, again 60. That is true because as you move to the right in the 12s row, each new number is 12 more than the previous number, since these are the successive multiples of 12.

So this 48 in the 4th Ms column is in a sense “the magic number.” Whether you look below this 48 or you look to the right of it, you get a number that is exactly 12 more than 48. Those two numbers — the upper being a multiple of 12, the lower being a multiple of 15 — therefore must be the same. And they are. They are both 60.

So 60 must be a common multiple for 12 and 15. Not only that, but since this is the first time this happens in the chart, 60 is the Least Common Multiple, or LCM. Here is an updated chart that shows these relationships:

The general principle, as far as the chart goes, is this: when we find the column in which the difference equals smaller number, the multiple of the smaller number in that column is the “magic number.” When we look below this “magic number” and to the right of it, we see the same number, and this number is the LCM. The reason, in general, is this: with a difference equal to the smaller number, the number below the “magic number” must be greater by the value of the smaller number. And the number to the right of the “magic number” must also be greater by the value of the smaller number because that row simply lists the multiples of the smaller number.

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

To make it easier, let’s look at one more example. This time we’re seeking the LCM for 20 and 24. So now the smaller number = 20; larger number = 24; gap = 4. Chart looks like this:

Scan the chart to see where the difference equals the value of the smaller number, 20. This happens in the 5th Ms column. So the “magic number” is 100, and looking below and to the right of 100, we find the LCM, 120.

You might be wondering, though, if it is possible to mentally figure out in which column the difference will equal the smaller number. For if we could do this, we could find the LCM WITHOUT a chart, through this technique.

Turns out there is an easy way to do this. Figuring it out relates to the way that the gap makes the difference slowly but steadily increase as we read the chart left to right.

Look at the bottom row to see how the differences increase as we move from left to right:  4 — 8 — 12 — 16 — 20 …

These differences grow by 4 with each new pair of multiples. And that’s because the gap is 4.

How then do we figure out when that difference will be 20? This is like a 4th grade word problem. We just divide the smaller number, 20, by 4, and we get 5. This means that for this problem, the difference will be 20 in the 5th Ms column. And of course this is the case.

And this holds true in general. To find out in which column the difference equals the original smaller number, just divide the smaller number by the gap. The quotient you get gives the column number where this occurs.

So the process of finding the LCM, using this technique, boils down to just two math thought steps.

1st)  Figure out in which column the difference will equal the smaller number.

2nd)  Find the value of the larger number in that column.

And here are the corresponding two math steps:

1st)  Noting the gap, divide the smaller number by the gap, and get the quotient. (The quotient tells us the column number where we find the “magic number.”)

3rd)  Multiply the quotient by the larger number. This product is the multiple of the larger number found just BELOW the “magic number.” This will be 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)  (Smaller #) ÷ (Gap)  =  16 ÷ 4 = 4, so Quotient = 4.

2nd)  (Quotient) x (Larger #) = 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)  (Smaller #) ÷ (Gap)  =  21 ÷ 7 = 3, so Quotient = 3.

2nd)  (Quotient) x (Larger #) = 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)  (Smaller #) ÷ (Gap)  =  60 ÷ 12 = 5, so Quotient = 5.

2nd)  (Quotient) x (Larger #) = 5 x 72 = 360 = LCM.

Answer:  LCM for 60 and 72 is 360.

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PRACTICE:

Find the LCM for these pairs of numbers using the two-step process shown above.

a)   12 and 18

b)   15 and 20

c)   18 and 20

d)   24 and 28

e)   12 and 14

f)     14 and 21

g)   30 and 36

h)   36 and 45

i)     24 and 32

j)     36 and 48

a)   12 and 18; LCM  =   36

b)   15 and 20;  LCM  =  60

c)   18 and 20;  LCM  =  180

d)   24 and 28;  LCM  =  168

e)   12 and 14;  LCM  =  84

f)     14 and 21;  LCM  =  42

g)   30 and 36;  LCM  =  180

h)   36 and 45;  LCM  =  180

i)     24 and 32;  LCM  =  96

j)     36 and 48;  LCM  =  144

FINAL NOTE:  I should mention before signing off today that there are two types of situations 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. And there’s another situation in which we get the LCM in an even easier way. Fortunately, though, this is not terribly complicated. And I’ll explain these two situations in tomorrow’s post.

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