Common Factors

A factor of a number is an exact divisor of the given number. Every factor of a number is less than or equal to the given number, i.e. it cannot be greater than the given number. Every number has at least two factors, some numbers have more than two factors. For example, 1, 2, 3, 6 are the factors of 6. Also, 1 is a factor of every number and every number is a factor of itself. It can be said that the number of factors of a given number is finite. Also, check the highest common factor for any number here.

Common Factors Definition

In Maths, common factors are defined as factors that are common to two or more numbers. In other words, a common factor is a number with which a set of two or more numbers will be divided exactly.

Common Factors of Two Numbers

To find common factors of two numbers, first, list out all the factors of two numbers separately and then compare them. Now write the factors which are common to both the numbers. These factors are called common factors of given two numbers.

How to Find Common Factors

As we know, the factors are the numbers which divide the original number completely. But how to check if two or more numbers have common factors between them.

Follow the below steps to find the common factors.

  • Right the factors of the given numbers.
  • Find the common factor present in them.

Let us see some examples here.

Common factors of 15 and 25

Let us check the factors of the two numbers, i.e., 15 and 25.

15 = 1, 3, 5, 15

25 = 1, 5, 25

We can see, both 15 and 25 have 5 as the common factor.

Common Factors of 12 and 18

First, we need to write all the factors of 12 and 18.

Factors of 12 = 1,2,3,4,6, 12

Factors of 18 = 1,2,3,6,9, 18

Clearly we can see, the common factors between 12 and 18 are 1,2,3 and 6.

Common Factors of 8 and 24

Let us find the factors of 8 and 24.

Factors of 8 = 1,2,4,8

Factors of 24 = 1,2,3,4,6,8,12,24

So we can see here, the common factors of 8 and 24 are 1,2,4 and 8.

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Common Factors Examples

Understand more about common factors with the below examples.

Example 1: Find the common factors of 36 and 63.

Solution:

1 × 36 = 36

2 × 18 = 36

3 × 12 = 36

4 × 9 = 36

6 × 6 = 36

Stop here, since the number 6 is repeated.

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

1 × 63 = 63

3 × 21 = 63

7 × 9 = 63

9 × 7 = 63

Stop here, since the numbers 7 and 9 are repeated.

1, 3, 7, 9, 21 and 63 are factors of 63.

1, 3 and 9 are common in both the lists.

Hence, the common factors of 36 and 63 are 1, 3, 9.

We can also find the common factors for more than two numbers. Consider the below example to understand the process of finding common factors of three numbers.

Example 2: What are the common factors of 45, 80 and 28?

Solution:

1 × 45 = 45

3 × 15 = 45

5 × 9 = 45

9 × 5 = 45

Stop here, since the numbers 5 and 9 are repeated.

1, 3, 5, 9, 15 and 45 are factors of 45.

1 × 80 = 80

2 × 40 = 80

4 × 20 = 80

5 × 16 = 80

8 × 10 = 80

10 × 8 = 80

Stop here, since the numbers 8 and 10 are repeated.

1, 2, 4, 5, 8, 10, 16, 20, 40 and 80 are factors of 80.

1 × 28 = 28

2 × 14 = 28

4 × 7 = 28

7 × 4 = 28

Stop here, since the numbers 4 and 7 are repeated.

1, 2, 4, 7, 14 and 28 are factors of 28.

Only 1 is common in the above lists.

Hence, the common factor of 45, 80 and 28 is 1.

Applications

Common factors are used in solving problems like How to simplify fractions?

Consider the fraction 40/96.

40/96 = 5/ 12

Since, the common factors of 40 and 96 are 1, 2, 4, 8.

Select the greatest among them and express the numbers of the fraction as a multiple of this greatest number.

40 = 8 × 5

96 = 8 × 12

Other applications like comparing prices, understanding time-distance concepts and time & work problems.

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