Least common multiple helps a lot in many ways. The basic use of the least common multiple is to find factors of any equation. You must be thinking what is the least common multiple? It is the smallest multiple of two or more than two numbers. This number can't be zero because anything multiplied or divided by zero will always remain zero. This means that every number's LCM will always be equal to zero and that is why we exclude zero.

Let's create some examples for more clarity. For example, you need to find the LCM of 15 and 27. The objective is simple, we need to find the smallest common divisor that is common in both 15 and 27. If we break it into its factor it will be something like this:

15 = 3 \times 5

9 = 3 \times 3

Now we have the factors as well, we can find the LCM. The condition is that the number should be the smallest number as well as the common number. In the above example, if we multiply 3 \times 3 \times 5, it will be equal to 45. Hence, we can declare that 45 is the smallest number that can be divided by 15 and 9. This was pretty simple, right? This is because the numbers were easily broken into its factors but things get complicated as the number increases. That is why we came up with some steps that will help you to tackle any kind of question regarding LCM.

Calculating the Least Common Multiple

Calculating LCM isn't that hard. In fact, it is very easy once you understand it. Sometimes, you might be given more than 2 numbers and asked to find the LCM of those numbers. If you don't know how to find LCM then below are the steps to find LCM.

 

Q. Find the LCM of 72, 108, and 60.

 

Step No.1

The first thing is to decompose any number into its prime numbers. In the above example, we can break down the 72, 108, and 60 into their prime factors. Here is how you can decompose any number into its prime factors.

\begin{matrix} 2 \\ 2 \\ 2 \\ 3 \\ 3 \\  \\ \end{matrix}\begin{matrix} \left| 72 \right \\ \left| 36 \right \\ \left| 18 \right \\ \left| 09 \right \\ \left| 03 \right \\ \left| 01 \right \end{matrix} \qquad \begin{matrix} 2 \\ 2 \\ 3 \\ 3 \\ 3 \\  \\ \end{matrix}\begin{matrix} \left| 108 \right \\ \left| 054 \right \\ \left| 027 \right \\ \left| 009 \right \\ \left| 003 \right \\ \left| 001 \right \end{matrix} \qquad \begin{matrix} 2 \\ 2 \\ 3 \\ 5 \\ \\ \end{matrix}\begin{matrix} \left| 60 \right \\ \left| 30 \right \\ \left| 15 \right \\ \left| 05 \right \\ \left| 01 \right \end{matrix}

Step No.2

Once you decompose the numbers into their prime factors, now it's time to arrange them. This arrangement step isn't necessary but this step will make your work easy and a little messy.

72 = {2}^{3} \times {3}^{2}

108 = {2}^{2} \times {3}^{3}

60 = {2}^{2} \times 3 \times 5

Step No.3

This time, you need to take out the highest-powered number of each prime factor. In the above case, we have {2}^{3} in 72, {3}^{3} in 108, and 5 in 60. Now, multiply these numbers and the resulting number will be LCM.

{2}^{3} \times {3}^{3} \times 5 = 2,160

This concludes that 2,160 is the smallest number that can be divided by 72, 108, and 60. However, if a number is a multiple of another, then it is the LCM of both. For example, number 36 is a multiple of 12. Hence, we can say that 12 is the LCM of 36.

LCM (12, 36) = 36

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Relationship between the GCD and LCM

Although GCD and LCM are two different things, however at some points, they can be the same as well. The biggest difference is that the Greatest Common Divisor is the largest integer that can divide two or more than two numbers, on the other hand, Least common multiple is the smallest integer that can divide two or more than two numbers. Yet, you will find that GCD will be equal to LCM so don't get confused, it is normal. That is why we have an example in which you will see that LCM and GCM are same.

GCD (a, b) \qquad LCM (a, b)

GCD (12, 16) = 4

LCM (12, 16) = 48

48 \times 4 = 12 \times 16

192 = 192

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.