# Least Common Multiple

## What is LCM?

LCM stands for Least Common Multiple.

A multiple **is a number obtained by multiplying the given number by another whole number**.

When we consider two numbers, each will have its own set of multiples. Some multiples will be common to both numbers. The **smallest of these common multiples** is called the least common multiple (LCM) of the two numbers.

For example, multiples of 8 and 10 are,

8 - 8, 16, 24, 32, __40__

10 - 10, 20, 30, **40**

The common multiple is 40 and it’s the smallest. Hence the LCM of 8 & 10 is 40

## How to find the LCM of two or more numbers?

There are 3 methods to calculate the LCM of two numbers,

- by listing out the common multiples
- by prime factorization
- division method

### 1. By listing out the common multiples

**What is the LCM of 4 & 5?**

**Step 1** - List a few multiples of 4 & 5,

Multiples of 4 - 4, 8, 12, 16, 20, 24, 28

Multiples of 5 - 5, 10, 15, 20, 25, 30, 35

**Step 2** - Circle all the common multiples

**Step 3** - 20 is the number that is there in both the lists. It the smallest common number in the list

**Hence the LCM of 4 & 5 is 20**.

### 2. By prime factorization

## Let’s try and figure out LCM of 60 and 90 using the prime factorization.

**Step 1** - Represent the numbers in the prime factored form

60 = 2 x 2 x 3 x 5

90 = 2 x 3 x 3 x 5

**Step 2** - Multiply all the prime factors, however the common factors will be included only once.

LCM of 60 and 90 = 2 x 3 x 5 x 2 x 3 = 180

### 3. Division method

Let’s try and find LCM of 6 & 15 using the common division method

**Step 1** - Find a prime number which is a factor of at least one of the given numbers. Write this prime number on the left of the two numbers (as shown).

For each number in the right column, do the following:

**Step 2** - If the prime number in step 1 is a factor of the number, then divide the number by the prime and write the quotient below.

If the prime number in step 1 is **not a factor** of the number, then write the number in the row below as it is. Continue the steps until all co-prime numbers are left in the last row. Two numbers are said to be coprime if they share no common factors other than 1.

**Step 3** - The LCM is the product of all the prime numbers.

**LCM of 6 and 15 is 2 x 3 x 5 = 30**