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.

 

Multiplication Table of Numbers

Multiples of 1 Multiples of 2
Multiples of 3 Multiples of 4
Multiples of 5 Multiples of 6
Multiples of 7 Multiples of 8
Multiples of 9 Multiples of 10
Multiples of 11 Multiples of 12
Multiples of 13 Multiples of 14
Multiples of 15 Multiples of 16
Multiples of 17 Multiples of 18
Multiples of 19 Multiples of 20
Multiples of 21