How to find Common Factors?
Last Updated :
23 Jul, 2025
The factors that can divide any number one or more than one without leaving any remainder are termed as a common factor. After the factorization, when we compare the factors of two or more we will get that some of the factors are the same or common and those factors are known as common factors.
What are Common Factors?
When a number is divided by a divisor, that divisor is known as a factor of that number. A factor cannot be greater than the given number but can be less than or equal to the given number. But 1 is the common factor of every number and every number is a factor of itself. Therefore, if it is a multiple of that divisor then the number may be divided perfectly by many divisors.
For Example
The factors of
55 =1,5,11
45 = 1,3,5,9,15,45
here, 1 and 5 are the numbers that are perfectly dividing both the numbers. Thus the common factors of 55 and 45 are 1 and 5.
"When two or more numbers are exactly divided by the same number(s), then the common divisors of the given numbers are known as' common factors." A common factor is also defined as a number that divides two or more numbers exactly without leaving any remainder.
How to Find Common Factors
To find the common factor of two or more numbers, follow the steps below:
i) List the factors of each number.
ii) Compare the factors of every number.
iii) List the same (i.e. common) factors in every number. These factors are called common factors of the given numbers.
Greatest Common Factor
When the factorization of one or more numbers has been calculated then the greatest common factor of the given numbers is termed as The GCF-Greatest Common Factor. After that, there are some factors that are common in the given numbers. Out of these numbers, the number which is the largest is the greatest common factor.
Let's suppose that m and n are natural numbers.
The GCF of two natural numbers m and n is the largest possible number that divides m and n. It is also known as the HCF-Highest Common Factor or (GCD) greatest common divisor.
Example 1: Find the common factors of 48 and 68 and then find out the greatest common factor between them?
Solution:
Factors of 24 : 1, 2, 3, 4, 6, 8, 12, 24
Factors of 68 : 1, 2, 4, 17, 34 and 68.
Thus, the common factors of 24 and 68 are 1, 2, and 4. Out of these common factors, 4 is the Greatest common factor.
Therefore , the Greatest Common Factor of 24 and 68 is 4. It can be written as GCF(24, 68) = 4.
Example 2: Explain How to find the common factors?
Solution:
The factors are the numbers that are a number's exact divisors. There are some steps to take in order to identify the common factors.
Step 1 : Separately write down all the factors of the given numbers.
Step 2 : Now look for the factors that are common in the given numbers and write them down in a separate row.
Example : Find out the common factors of 8 and 16 ?
Solution : Factors of 8 are 1,2,4,8
Factors of 16 are 1,2, 4,8,16
so common factors of 8 and 16 are 1,2,4,8
Sample Problems
Problem 1: Find out the common factors of 17 and 21?
Solution:
The factors of 17 and 21 are,
Factors of 17 = 1, 17
1 x 17 = 17
17 x 1 = 17
Factors of 21 = 1,3, 7, 21
1 x 21 = 21
3 x 7 = 21
7 x 3 = 21
21 x 1 = 21
Thus, the common factor of 17 and 21 is 1.
Problem 2: Find the common factors of 56 and 36.
Solution:
The factors of 56 and 36 are,
Factors of 56 are
1 x 56 = 56
2 x 28 = 56
4 x 14 = 56
7 x 8 = 56
{ 1, 2, 4, 7, 8, 14, 28, 56 }
Similarly for the Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, and 36
Thus, the common factors of 56 and 36 are 1, 2, 4
Problem 3: Find the common factor of 16, 28, and 42. What will be the greatest common factor?
Solution:
The factors of the three numbers- 16, 28, and 42 are
Factors of 16 = 1, 2, 4, 8, and 16
Factors of 28 = 1, 2, 4, 7, 14, and 28
Factors of 42 = 1, 2, 3, 6, 7, 14, 21 and 42
The common factors of 16, 28, and 42 are 1, 2 . So, the greatest common factor of 16, 28, and 42 is 2 .
GCF(16,28,42) = 2
Problem 4: What are the common factors between 55 and 25? Find out the GCF?
Solution:
The factors of 55 are 1 , 5 , 11 and 55
The factors of 25 are 1, 5 and 25 .
Thus, the common factor of 55 and 25 are 1 and 5
here the greatest common factor is 5
GCF(55, 25) = 5
Problem 5: What are the common factors of 10 and 15?
Solution:
The factors of 10 are: 1, 2, 5, 10
The factors of 15 are: 1, 3, 5, 15
Therefore, the common factors of 10 and 15 are 1 and 5.
Problem 6: What are the common factors of 18 and 19? Find out the greatest common factor?
Solution:
The factors of 18 are : 1, 2, 3, 6, 9 and 18
The factors of 19 are : 1, 19
Therefore, the common factors of 18 and 19 are only 1 so it is only common factor or highest factor.
Practical Applications of Common Factors and GCF
The concept of common factors and the greatest common factor (GCF) has practical applications in various areas of mathematics, everyday problem-solving, and even in certain real-world situations. Some of the practical applications include
Simplifying Fractions
When simplifying fractions, the GCF of the numerator and denominator is used to reduce the fraction to its simplest form.
Factoring Polynomials
In algebra, the GCF is used to factor out the greatest common factor from polynomial expressions.
Problem Solving in Number Theory
The GCF is used in number theory problems, such as finding whether two numbers are coprime (if their GCF is 1).
Dividing Items into Groups
When dividing items into groups or making equal distributions, the GCF helps in determining the largest possible equal groups.
Simplifying Ratios
GCF is used to simplify ratios, which is important in various practical applications, such as cooking, construction, and financial planning.
Finding LCM (Least Common Multiple) Using GCF
The GCF used along with the product of two numbers to find the least common multiple (LCM). The formula for finding LCM is
LCM = Product of the numbers/ GCF
Reducing Exponents in Expressions
To simplify the expressions with exponents, the GCF helps to reduce the exponents.
Practice Problems
1.Find the common factors of the following pairs of numbers:
a) 12 and 18
b) 20 and 50
c) 45 and 60
2. Find the GCF of the following pairs of numbers:
a) 16 and 24
b) 36 and 48
c) 27 and 63
Related Questions:
What is the greatest common factor?
The Greatest Common Factor is the largest of the common factors of two or more numbers.
How to find common factors?
To find the common factor of two or more numbers,
i) List the factors of each number. ii) Compare the factors of every number. iii) List the same factors in every number.
Summary
The concepts of common factors and GCF are not just abstract mathematical ideas, but they are the tools that can be applied to make problem-solving more efficient and effective in various contexts.
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