Here you will learn concept of LCM and how to find least common multiple (LCM) of numbers and fractions with examples.

Let’s begin –

## Concept of LCM

Let \(n_1\) and \(n_2\) be two natural numbers distinct from each other. The smallest natural number n that is exactly divisible by \(n_1\) and \(n_2\) is called Least Common Multiple (LCM) of \(n_1\) and \(n_2\) and is designated as LCM(\(n_1\), \(n_2\)).

## How to Find Least Common Multiple (LCM)

(a) Find the standard form of the numbers.

(b) Write out all the prime factors, which are contained in the standard forms of either of the numbers.

(c) Raise each of the prime factors listed above to the highest of the powers in which it appears in the standard forms of the numbers.

(d) The product of results of the previous step will be the LCM of numbers.

**Note** : GCD(\(n_1\), \(n_2\)).LCM(\(n_1\), \(n_2\)) = \(n_1\).\(n_2\)

i.e. The product of the HCF and LCM equal to the product of the numbers.

## Rule for Finding LCM of Fractions

LCM of two or more fractions is given by :

\(LCM of Numerators\over HCF of Denominators\)

**Example** : Find the LCM of 150, 210, 375.

**Solution** : We have the numbers, 150, 210, 375.

1). Writing down the standard form of numbers.

150 = \(5 \times 5 \times 3 \times 2\)

210 = \(5 \times 2 \times 7 \times 3\)

375 = \(5 \times 5 \times 5 \times 3\)

2). Write down all the prime factors that appears at least once in any of the numbers : 5, 3, 2, 7

3). Raise each of the prime factors to their highest available power (considering each to the numbers).

Hence, the LCM will be \(2^1\times 3^1 \times 5^3 \times 7^1\) = 5250

**Example** : Find the LCM of 50, 75.

**Solution** : We have the numbers, 50, 75

1). Writing down the standard form of numbers.

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

75 = \(5 \times 5 \times 3\)

2). Write down all the prime factors that appears at least once in any of the numbers : 5, 3, 2

3). Raise each of the prime factors to their highest available power (considering each to the numbers).

Hence, the LCM will be \(5^2\times 3^1 \times 2^1\) = 150