What is GCF and what are the methods to find it?

What is GCF

GCF is the shortest form of Greatest Common Factor. It is the study of finding the factors that exactly divide the number. This is a number theory topic. GCF is widely used in number theory and modular arithmetic. In this post, we learn about the definitions, rules of GCF. We also study the common factors.

What is a Common Factor? 

A number that divides another number with no remainder is known as the factor of that number.

Factors are the numbers we can multiply together to get another number. When we find the factors of two or more numbers, there are some same factors that both the numbers have known as the common factors.

Example 1

Find the common factors of 12 and 16.

Solution 

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

Factors of 16 = 1, 2, 4, 8, 16

Common factors of 12 and 16 = 1, 2, 4

Example 2

Find the common factors of 15 and 36.

Solution 

Factors of 15 = 1, 3, 5, 15

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

Common factors of 15 and 36 = 1, 3

Example 3

Find the common factors of 25 and 45.

Solution 

Factors of 25 = 1, 5, 25

Factors of 45 = 1, 3, 5, 9, 15, 45

Common factors of 15 and 36 = 1, 5

Example 4

Find the common factors of 24 and 30.

Solution 

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

Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30 

Common factors of 24 and 30 = 1, 2, 3, 6

Example 5

Find the common factors of 44 and 55.

Solution 

Factors of 44 = 1, 2, 4, 11, 22, 44

Factors of 55 = 1, 5, 11, 55 

Common factors of 44 and 55 = 1, 11

What is GCF?

GCF stands for Greatest Common Factor. The largest number that divides two or more numbers exactly is known as the greatest common factor. It is actually the number that comes when we find the factors of two or more numbers and then collect the common factors among them and at last find the largest number from common factors that largest number is known as GCF. GCF is also known as GCD (Greatest Common Divisor) or HCF (Highest Common Factor). To calculate the GCF with steps, use an online GCF Calculator.

How to Find GCF?

We can calculate the GCF in three ways

  1. By List of factors
  2. By Prime factorization 
  3. By Division method

By List of Factors

In this method find all the factors of two or more numbers then pick the common factors and select the largest one from the common factors this largest one is known as GCF. This method is applicable or quicker when you are dealing with smaller numbers. 

Example 1

Find the GCF of 12 and 16.

Solution 

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

Factors of 16 = 1, 2, 4, 8, 16

Common factors of 12 and 16 = 1, 2, 4

GCF of 12 and 16 = 4

Example 2

Find the GCF of 15 and 36.

Solution 

Factors of 15 = 1, 3, 5, 15

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

Common factors of 15 and 36 = 1, 3

GCF of 15 and 36 = 3

Example 3

Find the GCF of 25 and 45.

Solution 

Factors of 25 = 1, 5, 25

Factors of 45 = 1, 3, 5, 9, 15, 45

Common factors of 15 and 36 = 1, 5

GCF of 15 and 36 = 5

Example 4

Find the GCF of 24 and 30.

Solution 

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

Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30 

Common factors of 24 and 30 = 1, 2, 3, 6

GCF of 24 and 30 = 6

Example 5

Find the GCF of 44 and 55.

Solution 

Factors of 44 = 1, 2, 4, 11, 22, 44

Factors of 55 = 1, 5, 11, 55 

Common factors of 44 and 55 = 1, 11

GCF of 44 and 55 = 11

By Prime Factorization

Prime factors should be in prime numbers, prime numbers or coprime are those numbers that are only divisible by one or themselves.

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This method is applicable for larger numbers as a list of factors of a larger number is difficult. Prime factorization is a time-consuming method. Prime factorization is used to find the GCF of the set of numbers. Prime factorization follows three steps to calculate the GCF.

  1. Find the prime factors of each number
  2. Circle every common prime factor 
  3. Multiply all the circled numbers

The result is the GCF.

Example 1

Find the GCF of 60 and 90 by using prime factorization.

Solution

prime factorization

Prime factorization of 60 = 2x2x3x5

Prime factorization of 90 =2x3x3x5

Common prime factors = 2x3x5

GCF = 2x3x5=30

Example 2

Find the GCF of 24 and 42 by using prime factorization

Solution 

Prime factorization of 24 = 2x2x2x3

Prime factorization of 42 = 2x3x7

Common prime factors = 2×3

GCF = 2×3 = 6

Example 3

Find the GCF of 144 and 121 by using prime factorization

Solution 

Prime factorization of 144 = 2x2x2x2x3x3

Prime factorization of 121 =11×11

Common prime factors =no common factors

GCF = 1

Example 4

Find the GCF of 85 and 145 by using prime factorization

Solution 

Prime factorization of 85 = 5×17

Prime factorization of 145 = 5×29

Common prime factors = 25

GCF = 5

By Division method

For a large number of values, we use this method in which we follow long division. It follows the steps

  1. Divide the greatest number by the smallest number
  2. If the remainder is zero then the divisor is GCF. If the remainder is not zero, then make the remainder of the above step as the divisor and the divisor of the above step as dividend and continue this process until the remainder becomes zero.

Example 1

Find the GCF of 198 and 360 by using the division method.

Solution

Take 198 as divisor and 360 as dividend

division method

Hence the GCF= 18

Example 2

Find the GCF of 126 and 162 by using the division method.

Solution

Take 126 as divisor and 162 as dividend

GCF

GCF = 18

GCF can also be calculated using the online GCF calculator. This tool offers all of the methods to find GCF that we have discussed above.

Summary

After reading this article you will be able to find the common factor, GCF by a list of factors, prime factorization, and division method. Once you learn all the basics of GCF you will master it and nothing remain difficult after practicing.

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