Multiples - Common Multiples And Common Factors

In Maths, we have learned different concepts in primary and classes. Factors and Multiples are important concepts, especially working with reducing and expanding fractions. It is also used while finding the patterns in numbers. In this article, let us discuss the definition of multiples, common multiples, the difference between factors, and multiples with many solved examples.

Table of Contents:

What is Multiple?

A multiple is a result that is obtained by multiplying a number by an integer (it shouldn’t be a function). The multiples of the whole number are obtained by taking out the product of the counting numbers and that of whole numbers. For example, To find the multiples of 6 we multiply 6 by 1, 6 by 2, 6 by 3, and so on. The multiples are the product of this multiplication.

Example 1: Find the multiples of whole number 3
Multiplication: 3 x 1 3 x 2 3 x 3 3 x 4 3 x 5 3 x 6 3 x 7 3 x 8
Multiples of 3: 3 6 9 12 15 18 21 24
Solution: The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24 …
Example 2: Find the multiples of whole number 8
Multiplication: 8 x 1 8 x 2 8 x 3 8 x 4 8 x 5 8 x 6 8 x 7 8 x 8
Multiples of 8: 8 16 24 32 40 48 56 64
Solution: The multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64 …

Common Multiples

Multiples that are common to two numbers are known as common multiples of those numbers. Let us understand with the help of an example.

Consider two numbers– 30 and 45. Multiples of 30 and 45 are –

30 = 30, 60, 90, 120, 150, 180, 210, 240, 270…..

45 = 45, 90, 135, 190, 225, 270……….

We see that 90 and 270 are first two common multiples of 30 and 45. But what can be the real life use of common multiples?

Suppose Joe and Sam are running on a circular track. They start from the same point but Joe takes 30 second to cover a lap while Sam takes 45 second to cover the lap. So when will be the first time they meet again at the starting point?

This can be deduced from the list of common multiples. Sam and Joe will meet again after 90 minutes.

Common Factors

Factors that are common to two or more numbers are known as their common factors. So how to find the common factors? Consider two numbers A and B. Write down all the factors of A and B separately and observe the numbers which are common to both A and B. Let us understand the concept of common factors with the help of an example.

Considering the same two numbers 30 and 45.

Multiples

Factors of 30 and 45 are –

30 = {1, 2, 3, 5, 10, 15, and 30}

45 = {1, 3, 5, 9, 15, 45}

What are the common factors that you can observe? 15, 5, 3 and 1 appears in both 30 and 45.

Common factors of 30 and 45 are 1, 3, 5 and 15.

This concept has lots of practical application too. Suppose you want to floor a room of dimension 30 m x 45 m. The maximum size of a square tile that can be used can be deduced with the help of common factors. As 15 is the highest common factor, a square tile of side length 15 cm should be used.

Co-prime Number

Two numbers having only ‘1’ as their common factor are known as co-prime numbers. Example: 5 and 14, 6, and 17.

Note:

  • Two prime numbers are always co-prime numbers.
  • Two even numbers can never be co-prime numbers.

Difference Between Factors and Multiples

The major difference between factors and multiples are provided below:

Factors Multiples
A factor of a number is defined as an exact divisor of the given number. A multiple of a number is defined as a number which is obtained by multiplying it by a natural number.
For example, the factors of 20 are 1, 2, 4, 5, 10, and 20. For example, the multiples of 20 are 20, 40, 60, 80, 100, etc

Multiples Examples

Example 1

Write the first 5 multiples of each of the following.

  1. 23 (b) 40  

Solution:

(a)The first five multiples of 23 are 23, 46, 69, 92, 115.

Explanation:

1 x  23 = 23

2 x 23 = 46

3 x 23 = 69

4 x 23 = 92

5 x 23= 115

(b) The first five multiples of 40 are 40, 80, 120, 160, 200

Explanation: 

1 x 40 = 40

2 x 40 = 80

3 x 40 = 120

4 x 40 = 160

5 x 40 = 200

Example 2:

Write down the factors and multiples of 25.

Solution:

The factors of 25 are 1, 5 and 25

The multiples of 25 are 25, 50, 75, 100, 125, 150, and so on.

Frequently Asked Questions on Multiples

What are multiples?

The multiples of a number are defined as a number that is obtained as the product of two numbers.

Are multiples and products the same?

When we multiply two numbers, the answer is called the product. We can say the product is also called the multiples of two numbers.

Are the multiples of 4 even?

Yes, the multiples of 4 are even. Since 4 is an even number, the multiples of 4 are also even.

Are the multiples of 3 always the multiples of 6?

Yes, the multiples of 3 are the multiples of 6, because 3 is a factor of 6.

What are the multiples of 7?

The multiples of 7 are 7, 14, 21, 28, 35, 42, and so on.

To learn more about factors and multiples, LCM and HCF, download BYJU’S- The Learning App.

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