Least Common Multiple

Least Common Multiple
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The least common multiple of a number is the smallest number that is the product of two or more numbers. While you process that definition, you will also be introduced to the least common multiple definition, LCM examples and how to find least common multiple. Check-out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions on Least Common Multiple at the end of the page. 

In this video, play around with fraction rods. Using them, learn about multiples and how to smartly use it for calculating the LCM in simple steps.

Lesson Plan

What is Least Common Multiple (LCM)?

The least common multiple of two or more numbers is the smallest number among all common multiples of the given numbers.

Let’s take two numbers — say, 2 and 5

Each will have its own set of multiples. 

  • Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …
  • Multiples of 5 are 5, 10, 15, 20, …

Representing these multiples on the number line and circling the common multiples,

multiples of 2 and 5

The common multiples of 2 and 5 are 10, 20, ….

The smallest number among 10, 20, … is 10

LCM of 2 and 5

So the least common multiple of 2 and 5 is 10

This can be written as:

 \(\therefore\) LCM (2, 5) = 10

The least common multiple is also known as "LCM" (or) the "Lowest Common Multiple" in Math. 


How to find the Least Common Multiple?

There are 3 methods to find the least common multiple of two numbers:

Each method is explained below with some examples of LCM.

1. By Listing Out the Common Multiples

Let’s learn this method using the example below.

Example:

Find the least common multiple (LCM) of 4 and 5.

Solution:

Step 1 - List a few multiples of 4 and 5 

  • Multiples of 4 - 4, 8, 12, 16, 20, 24, 28
  • Multiples of 5 - 5, 10, 15, 20, 25 

Step 2 - Circle all the common multiples.

LCM of 4 and 5

Step 3 - 20 is the first (smallest) number that is present in both the lists.

Hence by the definition of least common multiple, the LCM of 4 and 5 is 20

LCM of 4 and 5 is 20

2. By Prime Factorization

Let’s learn this method using the examples below.

Example 1

Find the least common multiple (LCM) of 60 and 90 using prime factorization.

Solution

Step 1 - Represent the numbers in the prime factored form.

 

prime factorisation of 60

prime factorisation of 90

\[\begin{align} 60&= 2 \times 2 \times 3 \times 5\\ 90&=2 \times 3\times 3 \times 5 \end{align} \]

Step 2 - The LCM of the given two numbers is the product of all the prime factors. However, common factors will be included only once. 

prime factorisation of 60 and 90

Still confusing? Then let’s see the following Venn diagram to understand this process easier.

venn diagram of multiples

Thus, LCM of 60 and 90 = \(2 \times 2 \times 3 \times 5 \times 3\) = 180

 ∴ LCM of 60 and 90 is 180

Example 2

Find the least common multiple (LCM) of 24 and 60 using prime factorization.

Solution

Step 1 - Represent the numbers in the prime factored form

prime factorisation of 24

prime factorisation of 60

Thus,
\[ \begin{array}{l}
24=2^{3} \times 3^{1} \\
60=2^{2} \times 3^{1} \times 5^{1}
\end{array} \]

Step 2 - The LCM of the two numbers will be the product of the greatest power of each prime factor occurring in both the numbers.
\[ \mathrm{LCM}(24,60)=2^{3} \times 3^{1} \times 5^{1}=120 \]

∴ LCM of 24 and 60 is 120

3. By Division Method

Let’s learn this method using the example below.

Example

Find the least common multiple (LCM) of 6 and 15 using the common division method.

Solution

Step 1 - Find a prime number which is a factor of at least one of the given numbers.

Write this prime number on the left of the two numbers (as shown).

LCM of 6 and 15

For each number in the right column, do the following: 

Step 2 - If the prime number in step 1 is a factor of the number, then divide the number by the prime and write the quotient below. 

If the prime number in step 1 is not a factor of the number, then write the number in the row below as it is. 

Continue the steps until all coprime numbers are left in the last row. 

Two numbers are said to be coprime if they share no common factors other than 1.

LCM of by division method

Step 3 - The LCM is the product of all the prime numbers. 

LCM of 6 and 15 is \(2 \times 3\times 5 = 30\)

 ∴ LCM of 6 and 15 is 30

Though we have three methods to find the LCM, the prime factorization method is the most common and easy method that we use.


Least Common Multiple Calculator

Let’s enter two numbers in the calculator below and it would give the LCM of the given numbers along with a step-by-step procedure using division method.

Using the calculator, we can check our LCM after solving using one of the above three methods.


 
Thinking out of the box
Think Tank
  1. What is the least common multiple of two prime numbers?
  2. Find the least number which when divided by 15, 18, and 21 leave a remainder of 4 in each case.

Solved Examples

Here you can find some examples of LCM.

Example 1

 

 

Find the LCM of 980 and 9000 using the prime factorization method.

Solution

First, we carry out the prime factorization of the two numbers:

prime factorisation of 980

prime factorisation of 9000

The LCM will be the product of the greatest power of each prime factor occurring in the numbers:

\[\begin{align}
  {\text{LCM}}\left( {980,\;9000} \right) &= {2^3} \times {3^2} \times {5^3} \times {7^2} \hfill \\
  \qquad \qquad \qquad \qquad \quad \; &= {441000} \hfill \\ 
\end{align} \]

 ∴ LCM (980, 9000) = 441000
Example 2

 

 

Find the LCM of 371250 and 29040 using the prime factorization method.

Solution

First, we carry out the prime factorization of the two numbers:

prime factorisation of  371250 and 29040

prime factorisation of 29040

The LCM will be the product of the greatest power of each prime factor occurring in the numbers:

\[\begin{align}
  {\text{LCM}}\left( {371250,\;29040} \right) &= {2^4} \times {3^3} \times {5^4} \times {11^2} \hfill \\
  \qquad \qquad \qquad \qquad \quad \; &= {32670000} \hfill \\ 
\end{align} \]

 ∴ LCM (371250, 29040) = 32670000
Example 3

 

 

Find the LCM of the three numbers 168, 252, and 288 using the prime factorization method.

Solution

First, we carry out the prime factorization of the three numbers:


prime factorisation of 168

prime factorisation of 252

prime factorisation of 288

The LCM will be the product of the greatest power of each prime factor occurring in the three numbers:

\[\begin{align}
  {\text{LCM}}\left( {168,\;252,\;288} \right) &= {2^5} \times {3^2} \times {7^1} \hfill \\
  \qquad \qquad \qquad \qquad \quad {\text{ }} &= {2016} \hfill \\ 
\end{align} \]

 ∴ LCM (168, 252, 288) = 2016
Example 4

 

 

Bus X comes to a stop every 15 minutes and bus Y comes to the same stop every 40 minutes.

Let us assume that both buses are currently at the same bus stop.

After how many hours will both the buses reach the same stop again?

Solution

To find the required time, we have to find the LCM of 15 and 40.

Multiples of 15 are 15, 30, 45, 60, 75, 90, 105, 120, 135, 150....

Multiples of 40 are 40, 80, 120, 160,...

So the LCM of 15 and 40 is 120.

Thus both buses reach the same stop again after 120 minutes, which is the same as 2 hours.

∴ Required time = 2 hours
 
Challenge your math skills
Challenging Questions
  1. Two positive numbers \(a\) and \(b\) are written as \[\begin{align} a = {x^3}{y^2} \textbf { & } b = x{y^3} \end{align}\] \(x\) and \(y\) are prime numbers. Find \({\text{LCM}}(a,b)\).
  2. The traffic lights at three different places change after every 52 seconds, 78 seconds, and 117 seconds respectively. If they change simultaneously at 7 AM, at what time will they change simultaneously again?

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about the Least Common Multiple with the simulations and practice questions. Now you will be able to easily understand the least common multiple definition, lcm examples, to find least common multiple and lowest common multiple of 5 and 10.

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Frequently Asked Questions (FAQs)

1. How do I find the least common multiple in math?

There are 3 methods to calculate the LCM of two numbers: by listing out the common multiples, by prime factorization method, and by division method.

These methods have been explained in the previous sections with examples.

2. What is the fastest way to find the LCM?

There are 3 methods to find the least common multiple of two numbers: 

  1. by listing out the common multiples 
  2. by prime factorization 
  3. by division method

Each method is explained in detail in the "How to find the Least Common Multiple?" section of this page.

If the numbers are smaller, we prefer methods 1 and 2

If the numbers are larger, we prefer method 3

3. What is the LCM of 12 and 9?

Few multiples of 12 are 12, 24, 36, 48, 60, 72, 84, ...

Few multiples of 9 are 9, 18, 27, 36, 45, …

The LCM of 4 and 9 is the smallest multiple that is a common multiple of both of the numbers.

Thus, the LCM of 12 and 9 is 36

4. What is the LCM of 4 and 10?

Few multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …

Few multiples of 10 are 10, 20, 30, 40, 50, ...

The LCM of 4 and 10 is the smallest multiple that is a common multiple of both of the numbers.

Thus, the LCM of 4 and 10 is 20

5. What is the LCM of 12 and 8?

Few multiples of 12 are 12, 24, 36, 48 …

Few multiples of 8 are 8, 16, 24, 32, …

The LCM of 12 and 8 is the smallest number among all common multiples of both of the numbers.

Thus, the LCM of 12 and 8 is 24

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