## The Least Common Multiple

Before we can dive into the concept of the Least Common Multiple (LCM), we must understand what a multiple is.

Multiple of a number is the product you get when you multiply that number with any integer. For example, multiples of $\hspace{0.2em} 2 \hspace{0.2em}$ include $\hspace{0.2em} 2 \hspace{0.2em}$, $\hspace{0.2em} 4 \hspace{0.2em}$, $\hspace{0.2em} 6 \hspace{0.2em}$, etc.

Now consider this.

Multiples of $\hspace{0.2em} 2 \hspace{0.2em}$ - $\hspace{0.2em} 2 \hspace{0.2em}$, $\hspace{0.2em} 4 \hspace{0.2em}$, $\hspace{0.2em} {\color{Red} 6} \hspace{0.2em}$, $\hspace{0.2em} 8 \hspace{0.2em}$, $\hspace{0.2em} 10 \hspace{0.2em}$, $\hspace{0.2em} {\color{Teal} 12} \hspace{0.2em}$, $\hspace{0.2em} 14 \hspace{0.2em}$, $\hspace{0.2em} 16 \hspace{0.2em}$, $\hspace{0.2em} {\color{DarkOrange} 18} \hspace{0.2em}$, $\hspace{0.2em} 20 \hspace{0.2em}$ ...

Multiples of $\hspace{0.2em} 3 \hspace{0.2em}$ - $\hspace{0.2em} 3 \hspace{0.2em}$, $\hspace{0.2em} {\color{Red} 6} \hspace{0.2em}$, $\hspace{0.2em} 9 \hspace{0.2em}$, $\hspace{0.2em} {\color{Teal} 12} \hspace{0.2em}$, $\hspace{0.2em} 15 \hspace{0.2em}$, $\hspace{0.2em} {\color{DarkOrange} 18} \hspace{0.2em}$, $\hspace{0.2em} 21 \hspace{0.2em}$ ...

See how some of the numbers are multiples of both $\hspace{0.2em} 2 \hspace{0.2em}$ and $\hspace{0.2em} 3 \hspace{0.2em}$? These ($\hspace{0.2em} 6 \hspace{0.2em}$, $\hspace{0.2em} 12 \hspace{0.2em}$, and $\hspace{0.2em} 18 \hspace{0.2em}$) are few of the common multiples of $\hspace{0.2em} 2 \hspace{0.2em}$ and $\hspace{0.2em} 3 \hspace{0.2em}$.

And because $\hspace{0.2em} 6 \hspace{0.2em}$ is the smallest of those common multiples, it is the least common multiple of $\hspace{0.2em} 2 \hspace{0.2em}$ and $\hspace{0.2em} 3 \hspace{0.2em}$.

The least common multiple (or LCM) of a group of numbers is the smallest number that is a multiple of each of the given numbers. In other words, the LCM is the least of their common multiples.

## How to Find the Least Common Multiple?

There are three popular methods of finding the LCM. Let's look into each of them one by one.

### Prime Factorization Method

Example

Find the LCM of $\hspace{0.2em} 12 \hspace{0.2em}$, $\hspace{0.2em} 15 \hspace{0.2em}$, and $\hspace{0.2em} 50 \hspace{0.2em}$.

Solution

Step 1. Do the prime factorization of each number – split them into their prime factors.

Step 2. Highlight the maximum occurrence of each of the prime factors present in any of the numbers.

So in this case, as you can see above, $\hspace{0.2em} 2 \hspace{0.2em}$ occurs twice in $\hspace{0.2em} 12 \hspace{0.2em}$, so we highlighted that. Similarly, $\hspace{0.2em} 5 \hspace{0.2em}$ occurs twice in $\hspace{0.2em} 50 \hspace{0.2em}$.

We highlight in only one of the two places (it doesn't matter which one), not both.

Also, make sure you don't miss any of the factors.

Step 3. Multiply the highlighted factors together to get the LCM.

Done!

Example

Find the LCM of $\hspace{0.2em} 12 \hspace{0.2em}$, $\hspace{0.2em} 45 \hspace{0.2em}$, and $\hspace{0.2em} 66 \hspace{0.2em}$.

Solution

Again, we do the prime factorization of each number. And then highlight the highest occurrence of each factor.

Finally, we multiply the highlighted factors to get the LCM. So,

A rather large LCM! But that's fine.

### Ladder Method

This is probably the most popular method of finding the LCM and is known by different names including the “box” method, “cake” method, or “grid” method.

Here's how it works.

Example

Find the LCM of $\hspace{0.2em} 5 \hspace{0.2em}$, $\hspace{0.2em} 6 \hspace{0.2em}$, and $\hspace{0.2em} 10 \hspace{0.2em}$.

Step 1. Write the numbers in a row and draw an L-shape around them.

Step 2. Try to find a prime number that can divide (evenly) into two or more of the given numbers. Write that prime factor on the left of the L.

Then divide the numbers and write the quotients below. If a number is not divisible, copy it down.

In the present example, $\hspace{0.2em} 2 \hspace{0.2em}$ is a common prime factor of both $\hspace{0.2em} 6 \hspace{0.2em}$ and $\hspace{0.2em} 10 \hspace{0.2em}$.

Step 3. Repeat the process with the new row(s) as long as you can find a prime factor common to two or more numbers in the row.

Step 4. When no two numbers in a row have any common factor, multiply the numbers on the left (the factors) and any numbers that remain in the last row. The product is your LCM.

Example

Find the LCM of $\hspace{0.2em} 12 \hspace{0.2em}$, $\hspace{0.2em} 24 \hspace{0.2em}$, and $\hspace{0.2em} 30 \hspace{0.2em}$.

Solution

Using the same steps from the previous example, let's create the ladder first.

Next, we multiply the factors and the remaining numbers (at the bottom) to get the LCM.

### The Mental Method

This is the method I use most often. If you are good with numbers and multiplication tables, you'd love it. It's simple and quick. And you can do it in your head.

Example

Find the LCM of $\hspace{0.2em} 5 \hspace{0.2em}$, $\hspace{0.2em} 6 \hspace{0.2em}$, and $\hspace{0.2em} 10 \hspace{0.2em}$.

Step 1. Pick the largest number. For this example, it would be 10.

Step 2. Go through the multiples of that number (from step $\hspace{0.2em} 1 \hspace{0.2em}$), starting with the smallest. And check if that multiple is divisible by the other numbers too (that would mean it's their multiple as well).

Step 3. The first multiple to pass this test is the LCM. So here, $\hspace{0.2em} 30 \hspace{0.2em}$ is our LCM.

Example

Find the LCM of $\hspace{0.2em} 4 \hspace{0.2em}$, $\hspace{0.2em} 6 \hspace{0.2em}$, and $\hspace{0.2em} 15 \hspace{0.2em}$.

Solution

Alright, in this example, the largest number is $\hspace{0.2em} 15 \hspace{0.2em}$. So, we go over its multiples one by one to see if they are also multiples of $\hspace{0.2em} 4 \hspace{0.2em}$ and $\hspace{0.2em} 6 \hspace{0.2em}$ (or divisible by $\hspace{0.2em} 4 \hspace{0.2em}$ and $\hspace{0.2em} 6 \hspace{0.2em}$).

$\hspace{0.2em} 60 \hspace{0.2em}$ is the first multiple that passes the test. So $\hspace{0.2em} 60 \hspace{0.2em}$ is the LCM of $\hspace{0.2em} 4 \hspace{0.2em}$, $\hspace{0.2em} 6 \hspace{0.2em}$, and $\hspace{0.2em} 15 \hspace{0.2em}$..

Well, that brings us to the end of this tutorial on what the least common multiple is and how to find it. Until next time.