LCM(Least Common Multiple)

Least Common Multiple

Multiple is a number that can be divided by the given number without leaving a reminder. For example:

 20 is a multiple of 5 

Or, 5 × 4 = 20

And, 20 ÷ 5 = 4

Least common multiple

The least common multiple of two numbers is the “smallest non-zero common number” which is a multiple of both the numbers.

The different methods to find least common multiple of two or more numbers are:

• Using prime factorization
• Using repeated division
• Using multiples

1. LCM using prime factorization

In this method, a factorization tree for each given number is generated by listing the multiples of that number. The last branch of the tree has the least prime factors for that number.

For example, the factorization trees for 36 and 48 are generated as follows:

Figure: Prime factorization trees for the number 36 and 48

To find the LCM, pair the common multiples as shown. List them along with the remaining multiples.

LCM = 2 × 2 × 3 × 3 × 2 × 2

LCM = 144

2. LCM using repeated division

In this method, the given numbers are divided by the common divisors until there is no possible further division by the common number. The divisors and the remainders are multiplied together to obtain the LCM.

LCM =  2 × 2 × 3 × 4 × 3

LCM = 144

3. LCM using multiples

To find the LCM using multiples, list the multiples of the numbers in the table as shown. The least common multiple is the first common multiple for the given numbers. 

	1	2	3	4	5	6	7	8	9	10	11	12
36	36	72	108	144	180	216	252	288	324	360	396	432
48	48	96	144	192	240	288	336	384	432	480	528	576

For 36 and 48, the number 144 is the LCM.

Application

The dimension of using LCM of two numbers starts with basic math operations such as addition and subtraction on fractional numbers. In math problems where we pair two objects against each other, the LCM value is useful in optimizing the quantities of the given objects. Also, in computer science, the LCM of numbers helps design encoded messages using cryptography.