What is a Common factor in Math?

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A factor of a number is defined as the number which is an exact divisor of the given number. Each factor is less than or equal to the given number i.e. it can not be greater than the number whose factors are being calculated.

Every number in maths has minimum two factors, 1 and the number itself. However, a given number may have more than two factors and thus, the number of factors of a given number can be finite. For example, 1,2,4,8 are the factors of number 8.

In Maths, a common factor is defined as the factor which is common to two or more numbers. In other words, a whole number which will divide exactly the two or more numbers without leaving a remainder is termed as a common factor.

How to find common factors?

To find the common factors:

Step1: List out all the factors of the numbers for which we have to calculate common factor separately

Step2: Now, do the comparison to find out the factors which are common to these numbers.

The factors that appear in both the numbers are called common factors.

Let us solve a simple problem to better understand the concept.

Problem 1: Find out the common factors of 8 & 12.

Solution:

The factors of 8 are: 1, 2, 4 and 8

The factors of 12 are: 1, 2, 3, 4, 6 and 12

1, 2 and 4 appear in while factorizing both the terms.

So, the common factors of 8 & 12 are: 1, 2 and 4.

Problem2: Find out the common factors of 15 & 25.

Solution:

On factorizing 15 we get: 1, 3, 5 and 15

On factorizing 25 we get: 1, 5 and 25

So, the common factors of 15 & 25 are: 1 and 5.

Similarly, we can find common factors of any of the given two or more numbers.

Real World applications:

Common factors are very helpful in solving various problems especially when simplifying the fractions. These may also be very useful in various other applications such as work problems, understanding the time-distance concepts and most importantly comparing the prices.

Let us now understand how to simplify the fractions using common factors.

Problem: Simplify the fraction 20/40

Factors of 20 are : 1, 2, 4, 5, 10, 20

Factors of 40 are: 1, 2, 4, 5, 10, 20, 40

Since, the common factors of 20 & 40 are: 1, 2, 4, 5, 10, 20

Pick the highest common factor and express the numbers of the fraction as a multiple of this greatest number.

Numerator becomes: 20 = 20 × 1

Denominator becomes: 40 = 20 × 2

20 in the numerator and denominator gets cancelled out. So, the equation automatically gets simplifies into ½ .

Problem 3: Find out the common factors of 45, 80 and 28?

The factors of 45 are: 1, 3, 5, 9, 15 and 45

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