Methods to Find LCM of Three Numbers – Examples and FAQ

Methods to Find LCM of Three Numbers

There are five methods to find the LCM (Least Common Multiple)-

• Listing Multiples
• Prime Factorization
• Ladder Method
• Division Method
• Using the GCF (Greatest Common Factor)

We will learn about them one by one.

LCM of Three Numbers by Listing Multiples

Write down the multiples of each number until one of the multiples is present on all lists, which are known as common multiples. Now Find the smallest common multiple. This number is the required LCM.

Example: LCM of 2, 4, and 12.

Multiples of each number – 

2:  2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22

4:  4, 8, 12, 16, 20, 24, 28, 32, 36, 40

12:  12, 24, 36, 48, 60, 72,

The common multiple of three numbers is 12 (bold and underlined)

So, the required LCM = 12

LCM of Three Numbers by Prime Factorization

First, we do prime factorization of the given numbers. Then list all the prime numbers found as they occur most times for any given number. Now, multiply the list of prime factors together for finding the LCM.

Example: LCM of 4, 12 and 18. 

Prime factorization of each number – 

4 = 2 × 2

12 = 2 × 2 × 3

20 = 2 × 2 × 5

Now, look at the prime factorization of 4, 12 and 20. 

The prime number 2 occurs most often for two times. (2 × 2 is present in the prime factorization of each number)

The prime number 3 occurs most often for one time. (3 is present only once in the prime factorization of 12 only)

The prime number 5 occurs most often for one time. (5 is present only once in the prime factorization of 20 only)

Multiplying all the prime numbers as each occurs most often, we find 2 × 2 × 3 × 5 = 60.

So, the required LCM = 60.

LCM of Three Numbers by Ladder Method

Write down all three numbers in a row. Divide the numbers in the row with a prime number that evenly divides at least two or more numbers and write the result into the next row.

If any number in the layer or row is not divisible, just write it down as it is. Continue dividing rows by prime numbers until there are no more prime numbers that divide evenly two or more numbers.

Example:  LCM of 10, 12 and 15.

Multiply the numbers in the L shape (left column and bottom row), required LCM = 2 × 3 × 5 × 1 ×  2 × 1 = 60.

LCM of Three Numbers by Division Method

Write down all the numbers in a row. Divide the numbers with such a prime number which evenly divides at least one of the numbers, and write the result into the next row. If any number is not divisible, write it down as it is.

Keep dividing with prime numbers that divide at least one number until we get only 1’s in the final row.

Example: LCM of 10, 16 and 27.

Multiply the prime numbers in the first column.

So the required LCM = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 = 2160.

LCM of Three Numbers by GCF

Find the LCM of the first two numbers. Now, find the LCM of the result obtained and the third number.

Example:  We will find the LCM of 6, 10, 12

First, find the LCM of the first two numbers 6, and 10, by any method explained above, and we will get 30.

Now find the LCM of 30 and the third number 12, which will be 60.

Examples

1. Find the LCM of 8, 12 and 30 using the division method. 

Solution: 

After solving by division method, LCM = 2 × 2 × 2 × 3 × 5 = 120.

2. Find the LCM of 6, 8 and 21 using the prime factorization method.

Solution: Prime factorization of each number – 

6 = 2 × 3

8 = 2 × 2 × 2

21 = 3 × 7

Multiplying all the prime numbers as each occurs most often, we take:

2 × 2 × 2 × 3 × 7 = 168

So, the required LCM = 168.