What Are Common Factors? Definition, Examples, Methods and Practice Questions

What Are Common Factors?

Common factors are the numbers that divide two or more numbers exactly without leaving any remainder. In simple words, if a number is a factor of each given number, it is called a common factor.

Understanding common factors is one of the basic concepts of mathematics. It helps students solve questions related to fractions, algebra, ratios, and many other mathematical topics. Before learning the highest common factor method, students should clearly understand what is common factor and how to identify it.

Definition of Common Factors

A common factor is a number that is a factor of two or more numbers at the same time.

Example

Find the common factors of 12 and 18.

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

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

Common Factors:
1, 2, 3, 6

Therefore, the common factors of 12 and 18 are 1, 2, 3, and 6.

What Is Common Factor?

Many students ask, what is common factor?

A common factor is simply a number that divides all the given numbers exactly.

For example:

Numbers: 15 and 25

Factors of 15:
1, 3, 5, 15

Factors of 25:
1, 5, 25

Common Factors:
1 and 5

Steps to Find Common Factors

Follow these simple steps.

1. Write all factors of the first number.
2. Write all factors of the second number.
3. Compare both lists.
4. Write the factors that appear in both lists.

Example

Find the common factors of 16 and 24.

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

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

Common Factors:
1, 2, 4, 8

Highest Common Factor Method

The highest common factor method is used to find the greatest number that divides two or more numbers exactly.

There are different methods to find the Highest Common Factor (HCF).

Method 1: Listing Factors

Example:
Find the HCF of 20 and 30.

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

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

Common Factors:
1, 2, 5, 10

Highest Common Factor:
10

Method 2: Prime Factorization Method

Example:
Find the HCF of 24 and 36.

Prime factors of 24:
2 × 2 × 2 × 3

Prime factors of 36:
2 × 2 × 3 × 3

Common prime factors:
2 × 2 × 3

HCF = 12

This is one of the easiest and most accurate highest common factor methods.

How to Find Highest Common Factor

Students often ask how to find highest common factor.

You can follow these steps.

Using Listing Method

Write all factors of each number.

Select the largest factor that appears in every list.

Using Prime Factorization

Write each number as a product of prime factors.

Multiply only the common prime factors.

The result is the Highest Common Factor (HCF).

Examples

Example 1

Find the common factors of 8 and 20.

Factors of 8:
1, 2, 4, 8

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

Common Factors:
1, 2, 4

Example 2

Find the HCF of 18 and 30.

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

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

Common Factors:
1, 2, 3, 6

Highest Common Factor:
6

Example 3

Find the common factors of 21 and 35.

Factors of 21:
1, 3, 7, 21

Factors of 35:
1, 5, 7, 35

Common Factors:
1 and 7

Importance of Common Factors

Learning common factors helps students:

• Understand divisibility rules.
• Simplify fractions.
• Solve HCF questions.
• Solve ratio and proportion problems.
• Learn algebra more easily.
• Improve problem-solving skills.

Common Mistakes

Students should avoid these mistakes.

• Forgetting to list all factors.
• Including numbers that are not factors of both numbers.
• Choosing a larger number that is not common.
• Confusing factors with multiples.

Key Points

• A common factor divides all given numbers exactly.
• Every pair of numbers has at least one common factor, which is 1.
• The greatest common factor is called the Highest Common Factor (HCF).
• The HCF can be found by listing factors or using prime factorization.

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Short Notes on Common Factors

Definition: A common factor is a number that divides two or more numbers exactly.

Steps to Find Common Factors:

• Write all factors of each number.
• Compare the lists.
• Select the factors common to all numbers.

Highest Common Factor (HCF):
The greatest among the common factors is called the Highest Common Factor.

Methods to Find HCF:

• Listing Factors
• Prime Factorization

Example:
12 → 1, 2, 3, 4, 6, 12

18 → 1, 2, 3, 6, 9, 18

Common Factors = 1, 2, 3, 6

HCF = 6

MCQs on Common Factors

1. What is a common factor?

A. A number that divides only one number

B. A number greater than all numbers

C. A number that divides two or more numbers exactly

D. A prime number

Answer: C

2. Which is the highest common factor of 12 and 18?

A. 2

B. 3

C. 6

D. 12

Answer: C

3. Which of the following is a common factor of 15 and 20?

A. 2

B. 5

C. 10

D. 15

Answer: B

4. Which method is commonly used to find HCF?

A. Addition

B. Multiplication

C. Prime Factorization

D. Subtraction

Answer: C

5. The common factors of 8 and 12 are

A. 1, 2, 4

B. 2, 6

C. 4, 8

D. 3, 6

Answer: A

6. Every pair of numbers has at least which common factor?

A. 0

B. 2

C. 1

D. 5

Answer: C

7. The HCF of 16 and 24 is

A. 2

B. 4

C. 6

D. 8

Answer: D

8. Which number is a common factor of 21 and 35?

A. 3

B. 5

C. 7

D. 9

Answer: C

9. Common factors are always

A. Multiples

B. Divisors

C. Fractions

D. Decimals

Answer: B

10. Which method is easiest for small numbers?

A. Listing Factors

B. Graph Method

C. Estimation

D. Rounding

Answer: A

Worksheet / Assignment

Q1. Find the common factors of:

a) 10 and 15

b) 18 and 27

c) 24 and 30

d) 16 and 40

e) 28 and 42

Q2. Find the Highest Common Factor (HCF) using the listing method.

a) 12 and 20

b) 18 and 24

c) 30 and 45

d) 36 and 54

e) 42 and 56

Q3. Find the HCF using prime factorization.

a) 24 and 36

b) 48 and 72

c) 30 and 42

d) 54 and 90

e) 60 and 84

Q4. State whether the following statements are True or False.

1. Every common factor divides each given number exactly.
2. Every pair of numbers has at least one common factor.
3. HCF is always smaller than or equal to each given number.
4. Prime factorization can be used to find HCF.
5. Common factors and multiples are the same.

Q5. Solve the following word problems.

1. Two ropes are 24 m and 36 m long. They are to be cut into equal pieces of maximum possible length. Find the length of each piece.
2. A teacher has 30 red pencils and 45 blue pencils. She wants to make identical groups without any pencils left over. Find the maximum number of groups.
3. Two school bells ring every 18 minutes and 24 minutes. What is their Highest Common Factor?
4. Find the HCF of 54 and 72 using prime factorization.
5. Explain in your own words what is common factor with one example.

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