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Least Common Multiple

The least common multiple is also known as lowest common multiple. It is abbreviated as LCM. It plays a very important role in arithmetic. It is calculated for two or more numbers. Least common multiple of more than two numbers is the least number that is divisible by all the numbers for which one need to find least common multiple.

In other words, least common multiple of "n" positive integers a, b, c, ... is the smallest number x, such that when x is divided by each a, b, c, ..., there is no remainder left.

Methods of Finding least Common Multiple:

Given below are the methods for finding LCM.

Venn Diagram Method:

The steps involved in calculating LCM via this method are as follows:

  • Find prime factors of each number.
  • Draw a Venn diagram.
  • Write common factors in common areas.
  • The product of all the numbers so obtained is the required LCM.

Calculate the LCM of 48 and 180.


48 = 2 x 2 x 2 x 2 x 3
180 = 2 x 2 x 3 x 3 x 5
Venn diagram of above factors is as follows:

Least Common Multiple

Prime Factorization Method:

The steps for finding LCM using this method are as follows:
  • Find prime factors of each number.
  • Write them in the form of exponents.
  • Take out highest exponent terms and multiply them.
  • The product is the LCM of given numbers.

Find prime factors of 9, 18 and 24.


Prime factors are as follows:
9 = 3 x 3
18 = 2 x 3 x 3
24 = 2 x 2 x 2 x 3

In exponential form
9 = $3^{2}$
18 = 2 x $3^{2}$
24 = $2^{3}$ x 3

So, required LCM = $2^{3}$ x $3^{2}$
= 8 x 9 = 72

Grid Method:

Grid method involves following steps:
  • Construct a grid as we do in prime factorization method.
  • Write all the numbers in it separated by comma.
  • Choose a prime number by which at least two numbers are divisible.
  • Write the quotient down each number and write other numbers (which are not divisible) as they are.
  • Repeat this process until not getting two numbers divisible by a prime number.


Find the LCM of 15, 20 and 30.


Following grid is formed:

Least Common Multiple Definition

LCM = 5 x 3 x 2 x 2 = 60.

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