Greatest Common Factor of two or more numbers is a common factor of the numbers which is the greatest among all the common factors. Greatest Common Factor is also called as Highest Common Factor(HCF).

Consider the numbers 12 and 30.

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

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30

So, we get the common factors of 12 and 30 as 1, 2, 3, and 6. Among these, the number 6 is the greatest. So, 6 is called as the greatest common factor of 12 and 30.

The greatest common factor of two or more numbers can be defined as the highest factor common to both the numbers. It can be obtained by making a list of factors of the given numbers and finding the greatest one which is common to all.

For example, let us find the GCF of 32, 36 and 48.

These numbers can be factorized as follows:

32 = 2 x 2 x 2 x 2 x 2
36 = 2 x 2 x 3 x 3
48 = 2 x 2 x 2 x 2 x 3

Here, 2 is repeating twice in all the products of the numbers. Thus, the GCF of 32, 36 and 48 is 2 x 2 = 4.

There are a few steps to be followed in finding the GCF of the given numbers.

  1. Find all the factors of each given number.
  2. Find the common factors of the given number.
  3. The greatest of all the factors obtained in (Step 2) is the required GCF.

Examples of Finding the Greatest Common Factor:

Given below are some examples that explain how to find the greatest common factor.

Example 1:

Find the greatest common factor of 18, 24 and 36.

Solution:
Step 1:

Factors of 18 = 1, 2, 3, 6, 9 and 18,
Factors of 24 = 1, 2, 3, 4, 6, 8, 12 and 24
Factors of 36 = 1, 2, 3, 4, 6, 9,12, 18 and 36.

Step 2:

Common factors of 18, 24 and 36 = 1, 2, 3 and 6.

Step 3:

Therefore, the Greatest common factor of 18, 24 and 36 = 6.

Example 2:

Find the greatest common factor of 27, 54 and 81.

Solution:
Step 1:

Factors of 27 = 1, 3, 9 and 27,
Factors of 54 = 1, 2, 3, 6, 9, 18, 27 and 54
Factors of 81 = 1, 3, 9, 27 and 81.

Step 2:

Common factors of 27, 54 and 81= 1, 3, 9 and 27.

Step 3:

Therefore, the Greatest common factor of 27, 54 and 81 = 27.

Common factors of two or more numbers are the factors which are common to all the given numbers.

For example, the factors of number 12 are 1, 2, 3, 4, 6 and 12. The factors of number 30 are 1, 2, 3, 5, 6, 10, 15 and 30. There are some factors which are common to both 12 and 30. They are 1, 2, 3, 6. These are called as the common factors of 12 and 30.

Steps in Finding Common Factors:

  • Find the factors of the given numbers.
  • Find the factors which are common for the given numbers.

Examples in Finding Common Factors

Given below are some examples that explains how to find the common factors of given numbers.

Example 1:

Find the common factors of 6 and 8

Solution:

Factors of 6 = 1, 2, 3 and 6
Factors of 8 = 1, 2, 4 and 8

Therefore, the common factors of 6 and 8 = 1 and 2.
It can be observed that each common factor (1 and 2) divide the given numbers 6 and 8 exactly with the remainder = 0.

Example 2:

Find the common factors of 8, 12 and 16

Solution:

Factors of 8 = 1, 2, 4 and 8
Factors of 12 = 1, 2, 3, 4, 6 and 12
Factors of 16 = 1, 2, 4, 8 and 16

Therefore, common factors of 8, 12 and 16 = 1, 2 and 4.

We observe that each common factor (1, 2 and 4) divides the given numbers 8, 12 and 16 exactly with the remainder = 0.

Example 3:

Find the common factors of 14, 21 and 42

Solution:

Factors of 14 = 1, 2 and 7
Factors of 21 = 1, 3 and 7
Factors of 42 = 1, 2, 3, 6, 7,14, 21 and 42

Therefore, common factor of 14, 21 and 42 = 1and 7

We can observe that each common factor (1 and 7) divides the given number 14, 21 and 42 exactly with the remainder = 0.

When a common divisor of two, three or more numbers is found to be a number that will exactly divide each of the numbers it is referred to as the greatest common divisor. The greatest common divisor considered for two, three or more numbers would be the greatest number that will exactly dividing each of the non prime numbers.

Numbers prime to each other are such that have no common divisor and in some cases it is better known as the common divisor, the greatest common measure and the common measure.

Examples of Greatest Common Divisor

Given below are some examples on the Greatest Common Divisor

Example 1:

What is the greatest common divisor for 6 and 10?

Solution:

Steps for operations are as follows:

Step 1: Analyse the numbers and find out the common divisor
Step 2: 2 is found to be the common divisor
Step 3: Quotients 3 and 5 have no common divisor, so 2 is the common divisor

Given below are some practice problems on the greatest common factor.

Practice Problem 1:

Find the LCM of 12 and 18

Practice Problem 2:

Find the LCM of 40 and 24

Practice Problem 3:

Find the LCM of 3, 6 and 9