# Multiples of a Number

A multiple of a number is obtained by multiplying the number by an integer. Let us understand the concept of multiples from our real life examples. If a chocolate costs 10 Rs and you want to buy 5 pieces of it, then you have to pay 10×5=50 Rs. This calculation is made with the help of multiples. In this section, we will discuss about multiples of a number.

Let us learn what are multiples?

Also Learn: Factors of a Number

#### Definition of Multiples

A number is called a multiple of another number m if it is the product of m and an integer.

From the definition, it is clear that a number x is called a multiple of m if we have

x = m × y

for some integer y.

We know that 6=3×2. So 6 is a multiple of 3 as well as of 2.

#### Can Multiples be Negative?

Yes, multiples of a number can be negative. Note that -4=2×(-2). So -4 is a multiple of 2 which is negative.

#### How to find Multiples

To find the multiples of a number m, we need to find the multiplications below:

m×1 = m

m×2 = 2m

m×3 = 3m

m×4 = 4m

m×5 = 5m

$\vdots$   $\vdots$

So the multiples of m are m, 2m, 3m, 4m, 5m, 6m, 7m, …

#### Examples of Multiples

Multiples of 2: The multiples of 2 are given below:

2×1=2

2×2=4

2×3=6

2×4=8

2×5=10

So the first five multiples of 2 are 2, 4, 6, 8 and 10. In a similar way, we have:

• The first five multiples of 3 are 3, 6, 9, 12 and 15.

• The first five multiples of 4 are 4, 8, 12, 16 and 20.

• The first five multiples of 5 are 5, 10, 15, 20 and 25.

#### Set of Multiples

As the multiples of a number m is the product of m and an integer, so the multiples of m are obtained as follows:

m×n=mn

where n varies over the set of integers.

So the set of multiples of m can be written as follows:

$\{mn : n \text{ is an integer}\}$

#### Relation between Multiples and Factors

Let m, n and k be integers. Suppose that we have the relation m×n=k. In this case, we say that the number k is a multiple of both m and n.

By the definition of factors, we can also say that both m and n are factors of the number k. So the factors and the multiples of a number are strongly connected.

For example, 6=2×3.

Here 2 and 3 are factors of 6. On the other hand, 6 is a multiple of both 2 and 3.

#### Properties of Multiples

1. Multiples can be negative. We have seen above -4 is a multiple of 2.

2. Every number is always a multiple of the number itself. For example, 10 is a multiple of 10 as 10=10×1.

3. Every number is always a multiple of 1. For example, 10 is a multiple of 1 as 10=1×10.

4. As 0=m×0 for any number m, so we can say that 0 is a multiple of any integer.

5. The number of multiples of a number is infinite since the multiples are obtained by multiplying the number with an integer and the number of integers is infinite.

#### Common Multiples

Let x and y be two numbers. We know that the multiples of x are x, 2x, 3x, 4x, 5x, and so on. Similarly, the multiples of y are y, 2y, 3y, 4y, 5y, and so on. The common numbers between the set of multiples of x and y are called the common multiples of x and y.

Among the common multiples, the least one is called the least common multiple (lcm) of x and y.

For example, let x=2 and y=3. The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, … and the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, …

So the common multiples of 2 and 3 are: 6, 12, 18,…

∴ the least common multiple of 2 and 3 is 6. In other words, the lcm of 2 and 3 is 6. Mathematically, lcm(2,3)=12.