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
Table of Contents
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 are 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.
What are 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.
