How To Find LCM of Two Numbers – Explanation with Examples

How To Find LCM of Two Numbers

LCM of two numbers is the Least (smallest) Common Multiple of the given two numbers. We mainly use four methods to find the LCM of two numbers explained below.

1. Finding LCM by Listing Multiples Method

To find the LCM of two numbers using this method, we list the multiples of each given number until we find the first common multiple. This first common multiple will be the required answer. Let us see an example.

Example: Find LCM of 6 and 15 using the listing multiple method. 

Let us list the multiples of 6 and 15.

Multiples of 6 will be: 6, 12, 18, 24, 30, 36 … etc.

Multiples of 15 will be: 15, 30, 45, 60, 75, 90 … etc.

Here, the number 30 is the first common multiple of both 6 and 15. 

So, the LCM of 6, 15 is 30.

2. Finding LCM using Prime Factorization Method

Let us understand this by calculating the LCM of two numbers using this method. 

Example: Find the LCM of two numbers, 25 and 45, using prime factorization.

First, we list the prime factors of both the given numbers.

25 = 5 × 5

45 = 3 × 3 × 5

List all the prime numbers found as they occur most times for any given number. 

The occurrence of prime numbers:

• The prime number 3 occurs most often for two times. ( 3 × 3 is present in the prime factorization of 45)
• The prime number 5 occurs most often for two times. (5 is present only once in the prime factorization of 25 but twice in 45)

Now, multiply the list of prime factors together to find the required answer.

Multiplying all the prime numbers as each occurs most often, we find 3 × 3 × 5 × 5 = 225

So, the LCM of 25, 45 = 225

3. Finding LCM using Division Method

It is the most common method to find the LCM of two numbers or even more. The steps we will use in this method are – 

First, write down both the numbers in a row. 

Divide the row with the smallest prime number that wholly divides at least one of the numbers, and write the result into the next row.

If any number is not divisible by that prime number, write it down as it is in the next row. (You can see the third row in the below example where 9 was not divisible by 2, so we wrote 9 in the fourth row.)

Keep dividing as explained earlier, and when the last row is 1’s, we should stop.

Example: Let’s find the LCM of 24 and 36 using the division method.

Multiply the prime numbers in the first column.

So the required LCM will be 2 × 2 × 2 × 3 × 3 = 72.

4. Finding LCM Using GCF Method 

 We use this method only if the greatest common factor (GCF) of two numbers is given. The formula used to find the LCM of two numbers by this method is:

L.C.M. = a × b/ GCF(a,b)

For example, for 10 and 24, the GCF will be 2. So, the LCM = (10 × 24) / 2 = 240/2 = 120.

Examples 

1. What is the LCM (Least Common Multiple) of 18 and 24 by division method?

Solution:

For 18 and 24, the LCM = 2 × 2 × 2 × 3 × 3 = 72.

2. What is the LCM of 18 and 21? Use prime factorization.

Solution: Writing down the prime factors of each number,

18 = 2 × 3 × 3

21 = 3 × 7

The occurrence of Numbers:

2: one time

3: two times

7: one time

So, the LCM of 18, 21 = 2 × 3 × 3 × 7 = 126.

3. What is the LCM of 8, 12 using the listing method?

Solution: Listing the multiples – 

Multiples of 8 will be: 8, 16, 24, 32, 40, 48, 56, ….

Multiples of 12 will be: 12, 24, 36, 48, 60, ….

Here, the number 24 is the first common multiple of both 8 and 12. So, the LCM of 8, 12 is 24.