NCERT Class VI Mathematics for Low Vision and Blind Students.

What are the common multiples of 4 and 6? They are 12, 24, 36, ... . Which is the lowest of these? It is 12. We say that lowest common multiple of 4 and 6 is 12. It is the smallest number that both the numbers are factors of this number.

The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples.

What will be the LCM of 8 and 12? 4 and 9? 6 and 9?

Example 8

Find the LCM of 12 and 18.

Solution :
We know that common multiples of 12 and 18 are 36, 72, 108 etc.
The lowest of these is 36.

Let us see another method to find LCM of two numbers.

The prime factorisations of 12 and 18 are :
12 = 2 × 2 × 3;
18 = 2 × 3 × 3

In these prime factorisations, the maximum number of times the prime factor 2 occurs is two; this happens for 12. Similarly, the maximum number of times the factor 3 occurs is two; this happens for 18. The LCM of the two numbers is the product of the prime factors counted the maximum number of times they occur in any of the numbers.

Thus, in this case LCM = 2 × 2 × 3 × 3 = 36.

Example 9

Find the LCM of 24 and 90.

Solution :
The prime factorisations of 24 and 90 are:
24 = 2 × 2 × 2 × 3;
90 = 2 × 3 × 3 × 5

In these prime factorisations the maximum number of times the prime factor
2 occurs is three; this happens for 24. Similarly, the maximum number of times
the prime factor 3 occurs is two; this happens for 90.
The prime factor 5 occurs only once in 90.

Thus, LCM = (2 × 2 × 2) × (3 × 3) × 5 = 360

Example 10

Find the LCM of 40, 48 and 45.

Solution :
The prime factorisations of 40, 48 and 45 are;
40 = 2 × 2 × 2 × 5
48 = 2 × 2 × 2 × 2 × 3
45 = 3 × 3 × 5

The prime factor 2 appears maximum number of four times in the prime factorisation of 48,
the prime factor 3 occurs maximum number of two times in the prime factorisation of 45,
The prime factor 5 appears one time in the prime factorisations of 40 and 45, we take it only once.

Therefore, required LCM = (2 × 2 × 2 × 2)×(3 × 3) × 5 = 720

LCM can also be found in the following way :

Example 11

Find the LCM of 20, 25 and 30.

Solution :
We write the numbers as follows in a row :

So, LCM = 2 × 2 × 3 × 5 × 5.

(A) Divide by the least prime number which divides atleast one of the given numbers. Here, it is 2. The numbers like 25 are not divisible by 2 so they are written as such in the next row.
(B) Again divide by 2. Continue this till we have no multiples of 2.
(C) Divide by next prime number which is 3.
(D) Divide by next prime number which is 5.
(E) Again divide by 5.