Least Common Multiple

The least among all common multiples of two or more numbers is referred to as the Least Common Multiple or LCM. LCM of two or more numbers can be evaluated through the following ways:

1. Listing of multiples  

2. Prime Factorization 

Let us consider some examples to have a better understanding of these concepts:

An example using the listing of multiples :

Find the LCM of 3 and 4.

Solution:

List all the multiples of 3 and 4 and select the least among them.

Multiples of 3 are: 3, 6, 9, 12, 15, 18, 21,  24, 27, 30

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

The figure above denotes the three common multiples at the intersection of the two circles. Clearly, the least among them is 12. Hence, the LCM of 3 and 4 is 12.

An example using Prime Factorization:

Find the LCM of 12 and 14

Solution:

Let’s find the prime factors of 12 and 14 respectively, using the factor tree.

So,  12 = 2 \( \times \) 2 \( \times \) 3

And, 14 = 2 \( \times \) 7

Now, select each factor where it appears the greater number of times, so select 2, 2 and 3 from the factors of 12, and select 7 from the factors of 14. Do not Select 2 again from the factors of 14, as we have selected it from factors of 12.

Now, 

LCM of 12 and 14 is the product of the selected factors

LCM of 12 and 14 = 2 \( \times \) 2 \( \times \) 3 \( \times \) 7 

                            = 84

An examples of finding the LCM of more than two numbers:

Find the LCM of 15, 25 and 50:

Solution:

Prime Factors of 15, 25 and 50 are:

15 = 3 \( \times \) 5

3 appears only once here, so highlight it,

25 = 5 \( \times \) 5

5 appears most often here,

50 = 2 \( \times \) 5 \( \times \) 5

2 appears only once here, so highlight it, 

LCM = 2 \( \times \) 3 \( \times \) 5 \( \times \) 5

So the LCM of 15, 25 and 50 is 150.