LCM Caclulator (With Steps) — Least Common Multiple

This LCM calculator lets you calculate the LCM (least common multiple) of upto $\hspace{0.2em} 15 \hspace{0.2em}$ numbers at a time. The calculator will tell you not only the LCM but also how to calculate it using different methods, like prime factorization and ladder methods.

Here's a quick overview of what the LCM is and how to calculate it.

For those interested, we have a more comprehensive tutorial on the least common multiple.

What Is the LCM?

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.

For example, consider the numbers $\hspace{0.2em} 3 \hspace{0.2em}$ and $\hspace{0.2em} 5 \hspace{0.2em}$ .

Multiples of $\hspace{0.2em} 3 \hspace{0.2em}$ — $\hspace{0.2em} 3, 6, 9, 12, {\color{Red} 15} , 18, 21, 24, 27, {\color{Teal} 30} \hspace{0.2em}... \hspace{0.2em}$
Multiples of $\hspace{0.2em} 5 \hspace{0.2em}$ — $\hspace{0.2em} 5, 10, {\color{Red} 15} , 20, 25, {\color{Teal} 30} , 35 \hspace{0.2em}... \hspace{0.2em}$

As you can see, $\hspace{0.2em} {\color{Red} 15} \hspace{0.2em}$ and $\hspace{0.2em} {\color{Teal} 30} \hspace{0.2em}$ are two of the common multiples of $\hspace{0.2em} 3 \hspace{0.2em}$ and $\hspace{0.2em} 5 \hspace{0.2em}$ (there's an endless list of them as we go higher). And since $\hspace{0.2em} {\color{Red} 15} \hspace{0.2em}$ is the smallest of the common multiples, it is their least common multiple (lcm).

LCM Calculation

There are variuos different methods to calculate the LCM. One of the most popular methods is the one using prime factorization. Let me explain with an example.

Prime Factorization Method

Say, you want to find the LCM of $\hspace{0.2em} 36 \hspace{0.2em}$ , $\hspace{0.2em} 40 \hspace{0.2em}$ , and $\hspace{0.2em} 90 \hspace{0.2em}$ . Here's how you would go about it.

Step 1. Do the prime factorization of each of the numbers.

\begin{align*} 36 \hspace{0.2em} &= \hspace{0.2em} 2 \times 2 \times 3 \times 3 \\[1em] 40 \hspace{0.2em} &= \hspace{0.2em} 2 \times 2 \times 2 \times 5 \\[1em] 90 \hspace{0.2em} &= \hspace{0.2em} 2 \times 3 \times 3 \times 5 \end{align*}

Step 2. Identify the factors present in the prime factorization of the numbers. Note the instance of highest exponent (or maximum repititions) for each of those factors.

Here, the factors are $\hspace{0.2em} 2 \hspace{0.2em}$ , $\hspace{0.2em} 3 \hspace{0.2em}$ , and $\hspace{0.2em} 5 \hspace{0.2em}$ .

Also, the highest exponent of $\hspace{0.2em} 2 \hspace{0.2em}$ is $\hspace{0.2em} {\color{Red} 3} \hspace{0.2em}$ ( $\hspace{0.2em} 2 \hspace{0.2em}$ occurs $\hspace{0.2em} {\color{Red} 3} \hspace{0.2em}$ times in $\hspace{0.2em} 40 \hspace{0.2em}$ ). Similarly, the highest exponents for $\hspace{0.2em} 3 \hspace{0.2em}$ and $\hspace{0.2em} 5 \hspace{0.2em}$ are $\hspace{0.2em} {\color{Teal} 2} \hspace{0.2em}$ and $\hspace{0.2em} {\color{Orchid} 1} \hspace{0.2em}$ respectively.

Step 3. Multiply together the factors raised to their respective powers to get the LCM. So,

\begin{align*} \text{LCM} \hspace{0.3em} &= \hspace{0.25em} 2^ {\color{Red} 3} \times 3^ {\color{Teal} 2} \times 5^ {\color{Orchid} 1} \\[1em] &= \hspace{0.25em} 360 \end{align*}

LCM Calculator

About the LCM Calculator

Usage Guide

i. Valid Inputs

ii. Example

iii. Solutions

iv. Share

What Is the LCM?

LCM Calculation

Prime Factorization Method