LCM Calculator

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About the LCM Calculator

This LCM calculator lets you calculate the LCM (least common multiple) of upto 15\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.

Usage Guide

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i. Valid Inputs

You can enter a list of up to 15\hspace{0.2em} 15 \hspace{0.2em} positive integers (separated by commas) into the calculator.

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

iv. Share

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

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 3\hspace{0.2em} 3 \hspace{0.2em} and 5\hspace{0.2em} 5 \hspace{0.2em}.

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

As you can see, 15\hspace{0.2em} {\color{Red} 15} \hspace{0.2em} and 30\hspace{0.2em} {\color{Teal} 30} \hspace{0.2em} are two of the common multiples of 3\hspace{0.2em} 3 \hspace{0.2em} and 5\hspace{0.2em} 5 \hspace{0.2em} (there's an endless list of them as we go higher). And since 15\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 36\hspace{0.2em} 36 \hspace{0.2em}, 40\hspace{0.2em} 40 \hspace{0.2em}, and 90\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.

36=2×2×3×340=2×2×2×590=2×3×3×5\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 2\hspace{0.2em} 2 \hspace{0.2em}, 3\hspace{0.2em} 3 \hspace{0.2em}, and 5\hspace{0.2em} 5 \hspace{0.2em}.

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

LCM=23×32×51=360\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*}

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