Least Common Multiple

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Understanding LCM (Least Common Multiple)

Before we begin explaining about LCM, let us first get an idea of the concept with the help of this simple video:


Definition of Multiple

A multiple is a number that is obtained by multiplying the given number by another natural number.

For example:

Let’s consider \(3\).

Multiples of \(3\) are obtained by multiplying it by the natural numbers \(1, 2, 3, 4, 5, 6, …\).

So the multiples of \(3\) are \(3, 6, 9, 12, 15, 18, ...\)

multiples of 3 on number line

✍️Note: A number has an infinite number of multiples.


Definition of Common Multiple

A common multiple of two numbers is a number that is a multiple of each of the 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:

multiples of 2 and 5

Among these, the multiples that are common to each of these numbers are \(10, 20, ,...\). 

These are called the common multiples of \(2\) and \(5\).


Definition of 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.

In the previous example, 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:

 ∴ \(LCM (2, 5) = 10\)

LCM Examples

Using the above definition, the LCM of

  • \(3\) and \( 7\) is \(21\), i.e. LCM \((3,7) = 21\).
  • \(18\) and \(24\) is \(72\), i.e. LCM \((18,24) = 72\).
  • \(60\) and \(90\) is \(180\), i.e. LCM \((60,90) = 180\).
  • \(10\) and \(17\) is \(170\), i.e. LCM \((10,17) = 170\).

Methods of finding LCM

There are 3 methods to calculate the LCM of two numbers: 

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

1. By listing out the Common Multiples

Let’s learn this method using the example below.

Example:

What is the LCM of \(4\) & \(5\)?

Solution:

Step 1 - List a few multiples of \(4\) & \(5\), 

  • 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 the LCM of \(4\) & \(5\) is \(20\).

 ∴ \(LCM\) \(of\) \(4\) \(and\) \(5\) = \(20\)

2. By Prime Factorisation

Let’s learn this method using the examples below.

Example 1

What is the LCM of \(60\) and \(90\) using the prime factorization?

Solution

Step 1 - Represent the numbers in the prime factored form

prime factorisation of 60 and 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 \(90\) = \(2 \times 2 \times 3 \times 5 \times 3\) = \(180\).

 ∴ \(LCM\) \(of\) \(60\) \(and\) \(90\) = \(180\)

Example 2

What is the LCM of \(24\) and \(60\) using the prime factorization?

Solution

Step 1 - Represent the numbers in the prime factored form

prime factorisation of 24 and 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.
\[ \mathrm{LCM}(24,60)=2^{3} \times 3^{1} \times 5^{1}=120 \]

 ∴ \(LCM\) \(of\) \(24\) \(and\) \(60\) = \(120\)

3. Division Method

Let’s learn this method using the example below.

Example

What is the LCM of \(6\) & \(15\) 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 co-prime 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 \(2 \times 3\times 5 = 30\).

 ∴ \(LCM\) \(of\) \(6\) \(and\) \(15\) = \(30\)

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


Least Common Multiple Calculator

Let’s enter two or more numbers in the calculator below and it would give the LCM of the given numbers.

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


LCM Formula

Let's take any 2 numbers. If we know how to calculate the HCF of these 2 numbers, we can find their LCM using the formula given below:

\(\text{LCM of two numbers}= \frac{\text{Product of two numbers}}{\text{HCF of two numbers}}\)

Example:
Let’s find the LCM of \(6\) and \(15\) using the above formula.

Solution:
We know that the HCF of \(6\) and \(15\) is \(3\).
Using the above formula,
\[\begin{align}
\text{LCM }(6, 15)& = \dfrac{6 \times 15}{\text{HCF } (6, 15)}\\&= \dfrac{90}{3}\\&= 30
\end{align}\]

 ∴ \(LCM\) (6, 15)= \(30\)
 
Thinking out of the box
Think Tank
  1. How would you find the area of a sector of the circle?
  2. To warn ships of underwater rocks, a lighthouse spreads a red-coloured light over a sector of angle \({80^\circ}\). The light reaches a distance of 16.5 km. Find the area of the sea over which the ships are warned.
  3. Tom is riding a cycle to his friend's house. The wheel of his cycle has a radius of 20 cm and his friend lives 660 m away. How many times does his front wheel rotate when he goes and comes back from his friend's house?    

Solved Examples

Example 1

 

 

Find the LCM of \(980\) and \(9000\) using the prime factorisation method.

Solution:

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

prime factorisation of 980 and 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 factorisation method.

Solution:

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

prime factorisation of  371250 and 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 factorisation method.

Solution:

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


prime factorisation of 168, 252 and 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 as 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. Can you find the product of LCM and HCF of \(12\) and \(18\) without actually calculating their LCM and HCF? 
  3. Consider the numbers \(24\) and \(60\). Their HCF is \(12\). Find their LCM.
  4. Given that HCF \((306, 657) = 9\), find LCM \((306, 657)\).

Practice Questions

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

 
 
 
 
 
 

Frequently Asked Questions (FAQs)

1. How do you find the LCM?

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

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

2. What is an example of LCM?

You can refer to the sections “Methods of Finding LCM” and “Solved Examples” sections of this page for more examples of LCM.

3. What is the LCM of \(4\) and \(9\)?

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

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 \(4\) and \(9\) is \(36\).

4. What is the LCM of \(8\) and \(12\)?

Few multiples of \(8\) are \(8, 16, 24, 32, …\).

Few multiples of \(12\) are \(12, 24, 36, 48 …\).

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

Thus, the LCM of \(8\) and \(12\) is \(24\).

5. What is LCM in math?

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

 
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