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:

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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 \(3, 6, 9, 12, 15, 18, ...\)

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

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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:

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

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

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 \(10\).

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 \((10,17) = 170\).

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.

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

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

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. 

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

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

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).

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.

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.

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.

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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)= \(30\)