Relation Between Factors and Multiples

Understanding the relationship between factors and multiples is useful in solving many problems. These two are inversly related to each other through the multiplication and division operation. Let's learn each of these in detail and their relationship with each other with the help of examples.

Table of Contents

• Factors and Their Properties
• Multiples and Their Properties
• Relationship Between Factors and Multiples
• Solved Examples

Factors and Their Properties

Factors are numbers which divide a given number completely, leaving the remainder as zero.

Properties of Factors:
1. Every number is a factor of itself.
2. 1 is the factor of every number.
3. The factor of a number is always less than or equal to the number.
4. Every factor of a number divides it completely.
5. A number is always a factor of infinite numbers.
6. Every number has a few number of factors. The number of factors will not exceed the number.

Multiples and Their Properties

Multiples are numbers which are obtained by multiplying the given number with any number other than zero.

Properties of Multiples:
1. Every number is a multiple of each of its factors.
2. The multiple of a number is always greater than or equal to the number itself.
3. A number can be a multiple of more than one number.
4. The number of multiples of a number is infinite.

 

Relationship Between Factors and Multiples

The relationship between factors and multiples can be represented with a simple equation:
Factor A × Factor B = Multiple C
where ‘A’ and ‘B’ are the factors that multiply together to form multiple C. 

These two numbers can divide C evenly and C is the product of A and B. It can be divided evenly by both A and B. 
A factor is a "building block" of a number, while a multiple is the "result" of multiplying that number by an integer. 

Solved Examples on Factors and Multiples

Example 1. List the factors for the following numbers.
a. 36       
b. 27
c. 40
d. 48

Solution: 
a). Starting from 1, we need to check for all those numbers which divide 36 completely.

36 = 1 × 36
36 = 2 × 18
36 = 3 × 12
36 = 4 × 9 36 = 6 × 6

Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
 

b). Starting from 1, we need to check for all those numbers which divide 27 completely.

27 = 1 × 27
27 = 3 × 9
Factors of 27 are 1, 3, 9, 27.

 

c). Starting from 1, we need to check for all those numbers which divide 40 completely.

40 = 1 × 40
40 = 2 × 20
40 = 4 × 10
40 = 5 × 8
Factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40.

 

d. Starting from 1, we need to check for all those numbers which divide 48 completely.

48 = 1 × 48
48 = 2 × 24
48 = 3 × 16
48 = 4 × 12
48 = 6 × 8
Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.

 

Example 2. Write the first five multiples of the following.

a. 12  b. 17

Solution: 

a. Let’s list down all the multiples of 12:

12 × 1 = 12
12 × 2 = 24
12 × 3 = 36
12 × 4 = 48
12 × 5 = 60
The first five multiples of 12 are 12, 24, 36, 48, 60.

 

b. Let’s list down all the multiples of 17:

17 × 1 = 17
17 × 2 = 34
17 × 3 = 51
17 × 4 = 68
17 × 5 = 85

The first five multiples of 17 are 17, 34, 51, 68, 85.