Lowest Common Multiple 

Definition:

The Lowest (smallest or least) common multiple, of two or more numbers, are called the “Lowest Common Multiple” (LCM).

Example: Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30,……. and

Example: Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, …..

What is a “common multiple”?

There are 6, 12, 18, 24, 30,…. common multiples of 2 and 3.

Therefore, A common multiple, is a number that is a multiple of two or more numbers. 

Lowest common multiple of 2 and 3 is 6.

It is the smallest number in both multiples.

so, the L. C. M. of 2 and 3 = 6

Example: Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32,…….

Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48,…….

Lowest common multiple of 4 and 6 is 12.

             L. C. M. of 4 and 6 = 12

To find the L.C.M. of two or more numbers, we listed the first few multiples of numbers and stop when we found same multiples in both numbers.

Example: Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48,…….

Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72,…….

Lowest common multiple of 6 and 9 is 18.

             L. C. M. of 6 and 8 = 18

simply, it is the smallest of the common multiples.

Problems on Lowest common multiple

1.      Find the LCM of 12 and 18?

First Method:

Multiples of 12 are 12, 24, 36, 48, 60, 72, 84,…….

Multiples of 18 are 18, 36, 54, 72, 90, 108, 126,…….

Lowest common multiple of 12 and 18 is 36.

             L. C. M. of 12 and 18 = 36

Second Method:

The prime factorisations of 12 and 18 are

12 = 2 x 2 x 3

18 = 3 x 3 x 2

In these prime factorisations, 

The prime factor 2 comes maximum number of two times in the prime factorisation of 12, the prime factor three comes maximum number of two times in the prime factorisation of 18.

The LCM of the two numbers is the product of the prime factors counted the maximum number of times they comes in any of the numbers.

Therefore, LCM = 2 x 2 x 3 x

= 36.

2.      Find the LCM of 15, 20 and 25?

       We write the numbers as follows in a row,

                     So, LCM = 2 x 2 x 3 x 5 x 5 

                        LCM    = 300

         These are the steps to solve the LCM. 

Step 1. 

Divide by the least prime number which divides one of the given numbers.

Here, is the number is 2. Number 20 is divided by 2, and we get 10.

The numbers like 15 and 25 are not divisible by 2 so they are written as such in the next row.

Step 2.

Again divide by 2. Continue this till we have no multiples of 2.

Step 3.

Divide by next prime number which is 3.

Step 4.

Divide by next prime number which is 5

Step 5.

Again divide by 5.

3.     Find the LCM of 12, 18, 24 and 36?

          We write the numbers as follows in a row,

            Therefore, LCM = 2 x 2 x 2 x 3

LCM  = 72.