# LCM (Least Common Multiple)

## Introduction

LCM (Least Common Multiple or Lowest Common Multiple), they all are the same terms. We use them while finding out the number which is the lowest common multiple among many multiples of a number.
To see it in more detail how it is calculated, first of all, let’s review some general terms related to LCM.

## Multiple

A multiple is a number that we can get from a number when multiplied by any positive integer.

Example

What are the multiples of 7?
We can say there are unlimited number of multiples of 7.

To be short, 14 is a multiple of 7.
Why?
$$\because$$ here, 7 is the number
when 7 is multiplied by 2, the obtained number 14 is multiple 7.
How?
$${7} \times {2} = 14$$
Still, we can create more multiples of 7 by multiplying 7 with positive integers such as 3, 4, 5, 6, 7, 8….. and so on
$${7} \times {3} = 21$$
$${7} \times {4} = 28$$
$${7} \times {5} = 35$$
$${7} \times {6} = 42$$
$${7} \times {7} = 49$$
$${7} \times {8} = 56$$
So, here, 21, 28, 35, 42, 49, 56 are also multiples of 7.

## Common Multiples

Common multiples are calculated for two or more than two numbers. They are the multiples which are common among the multiples of the given numbers.

Example

Let’s understand it by taking the two numbers as 2 and 4 to find their common multiples.
Step1: find out the multiples of 2 and 4 separately.

Multiples of 2:
$${2} \times {1} = 2$$
$${2} \times {2} = 4$$
$${2} \times {3} = 6$$
$${2} \times {4} = 8$$
$${2} \times {5} = 10$$
$${2} \times {6} = 12$$
The multiples of 2 are 2, 4, 6, 8, 10, 12 and so on.
Multiples of 4:
$${4} \times {1} = 4$$
$${4} \times {2} = 8$$
$${4} \times {3} = 12$$
$${4} \times {4} = 16$$
$${4} \times {5} = 20$$
$${4} \times {6} = 24$$
The multiples of 4 are 4, 8, 12, 16, 20, 24 and so on.
Step2: Find out the common multiples of 2 and 4.
Finally, we can say 4, 8, 12 are the common multiples of 2 and 4, because the multiples 4, 8 and 12 do exist for both numbers 2 and 4.

## Least Common Multiple (LCM)

LCM (Least Common Multiple or Lowest Common Multiple) is the smallest number among the common multiples of given numbers.

Example

To understand LCM, lets reconsider the above example of common multiples, but now finding here the LCM of 2 and 4.
Find the LCM of 2 and 4.
So, what are the steps?
Step 1: Find out the multiples of 2.
Step 2: Find out the multiples of 4.
Step 3: Find out the common multiples of 2 and 4.
Step 4: Find out the smallest number among those common multiples, that will be the LCM of 2 and 4.
Step 1: Find out the multiples of 2.
$${2} \times {1} = 2$$
$${2} \times {2} = 4$$
$${2} \times {3} = 6$$
$${2} \times {4} = 8$$
$${2} \times {5} = 10$$
$${2} \times {6} = 12$$
The multiples of 2 are 2, 4, 6, 8, 10, 12 and so on.
Step 2: Find out the multiples of 4.
$${4} \times {1} = 4$$
$${4} \times {2} = 8$$
$${4} \times {3} = 12$$
$${4} \times {4} = 16$$
$${4} \times {5} = 20$$
$${4} \times {6} = 24$$
The multiples of 4 are 4, 8, 12, 16, 20, 24 and so on.
Step 3: Find out the common multiples of 2 and 4.
$$\therefore$$ the common multiples are 4, 8 and 12.
Step4: Find out the smallest number among those common multiples, that would be the LCM of 2 and 4.
$$\therefore$$ the smallest number among common multiples 4, 8 and 12 is 4.
Hence, LCM of 2 and 4 is 4.

## Relationship between LCM and HCF

### Product of LCM and HCF

Product of LCM and HCF of given numbers is equal to product of the numbers.

Formula

$$LCM \times HCF \; of \; given \; numbers = Product \; of \; the \; numbers$$

### Q) What is LCM?

HCF stands for Least Common Multiple or Lowest Common Multiple. It is the smallest number that is obtained from the common multiples of given number.

### Q) What are the methods to find LCM?

1) Prime factor method 2) Common division method 3) Common multiple method

### Q) What is the relationship between HCF and LCM?

For any numbers a and b: Product of their LCM and HCF = Product of the numbers

### Q) What is multiple?

Multiple is a value that we get by multiplying the number with any positive integer.

## Solved Examples

### 1) Find LCM of 12 and 24.

We can find LCM by common multiple method

Multiples of 12 are 12, 24, 36, 48, 60.

Multiples of 24 are 24, 48, 72, 96, 120.

From the above, the common multiples are 24, 48 and the lowest number of multiple is 24.

Therefore, LCM of 12 and 24 is 24.

### 2) Find LCM of 18 and 20 using prime factorization method.

Prime factorization of $$18 = 2 \times 3 \times 3 = 2 \times 3^2$$

Prime factorization of $$20 = 2 \times 2 \times 5 = 2^2 \times 5^1$$

From the above, common factors = 2.

To find the LCM, we take the highest power of common factor i.e. $$2^2$$

$$\therefore$$ LCM of 18 and 20 $$= 2^2 \times 3^2 \times 5^1$$

$$= 4 \times 9 \times 5$$

$$= 180$$

### 3) Find LCM of 18 and 20 using common division method.

Let's find LCM using common division method. LCM $$= 2 \times 2 \times 3 \times 3 \times 5$$

LCM $$= 4 \times 9 \times 5$$

$$= 180$$

### 4) Find LCM of 18 and 20 using common multiple method.

Multiples of 18 = 18, 36, 54, 72, 90, 108, 126, 144, 162

Multiples of 20 = 20, 40, 60, 80, 100, 120, 140, 160, 180, 200

Common multiple = 180

$$\therefore$$ LCM of 18 and 20 = 180

## Multiple Choice Questions

a) 12, 24, 36

b) 12, 18, 30

c) 6, 12, 18

d) 30, 36, 48

### 2) LCM of two coprime numbers x and y is

a) $$xy$$

b) $$\frac{x}{y}$$

c) $$x+y$$

d) $$x-y$$

### 3) LCM of two prime numbers a and b is

a) $$ab$$

b) $$\frac{a}{b}$$

c) $$a+b$$

d) $$a-b$$

a) 6

b) 3

c) 1

d) 2

a) 6

b) 2

c) 12

d) 4

a) 4

b) 5

c) 20

d) 1

a) 60

b) 3

c) 4

d) 5

a) 10

b) 5

c) 50

d) 1

a) True

b) False

c) Both

d) None of these

a) 60

b) 40

c) 24

d) 120