Common Factors
Common factors are factors that two or more numbers share. Finding common factors is a key step towards understanding the Highest Common Factor (HCF) and simplifying fractions.
In Class 4, students learn to list factors of two numbers and identify the ones that appear in both lists.
What is Common Factors - Class 4 Maths (Factors and Multiples)?
A common factor of two or more numbers is a number that is a factor of each of them.
For example, the factors of 12 are {1, 2, 3, 4, 6, 12} and the factors of 18 are {1, 2, 3, 6, 9, 18}. The common factors are the numbers in both lists: 1, 2, 3, 6.
Common factors of A and B = factors that divide both A and B exactly
Common Factors Formula
Steps to find common factors:
- List all factors of the first number.
- List all factors of the second number.
- Circle (or highlight) the factors that appear in both lists.
Solved Examples
Example 1: Example 1: Common Factors of 12 and 18
Problem: Find the common factors of 12 and 18.
Solution:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
Answer: Common factors of 12 and 18 = 1, 2, 3, 6
Example 2: Example 2: Common Factors of 8 and 20
Problem: Find the common factors of 8 and 20.
Solution:
Factors of 8: 1, 2, 4, 8
Factors of 20: 1, 2, 4, 5, 10, 20
Common factors: 1, 2, 4
Answer: Common factors of 8 and 20 = 1, 2, 4
Example 3: Example 3: Common Factors of 15 and 25
Problem: Find the common factors of 15 and 25.
Solution:
Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
Common factors: 1, 5
Answer: Common factors of 15 and 25 = 1, 5
Example 4: Example 4: Co-prime Numbers (Only 1 as Common Factor)
Problem: Find the common factors of 8 and 15.
Solution:
Factors of 8: 1, 2, 4, 8
Factors of 15: 1, 3, 5, 15
Common factor: 1 (only)
Answer: The only common factor is 1. Numbers with only 1 as a common factor are called co-prime numbers.
Example 5: Example 5: Using a Venn Diagram
Problem: Show the common factors of 16 and 24 using a table.
Solution:
Factors of 16: 1, 2, 4, 8, 16
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
| Only in 16 | Common | Only in 24 |
|---|---|---|
| 16 | 1, 2, 4, 8 | 3, 6, 12, 24 |
Answer: Common factors of 16 and 24 = 1, 2, 4, 8
Example 6: Example 6: Three Numbers
Problem: Find the common factors of 12, 18, and 24.
Solution:
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common to all three: 1, 2, 3, 6
Answer: Common factors = 1, 2, 3, 6
Example 7: Example 7: Word Problem (Equal Sharing)
Problem: Ria has 24 red beads and 36 blue beads. She wants to make identical necklaces using all beads, with each necklace having the same number of red and same number of blue beads. In how many ways can she do this?
Solution:
The number of necklaces must be a common factor of 24 and 36.
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12
Answer: She can make necklaces in 6 different ways (1, 2, 3, 4, 6, or 12 necklaces).
Example 8: Example 8: Consecutive Numbers
Problem: Find the common factors of 20 and 21.
Solution:
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 21: 1, 3, 7, 21
Common factor: only 1
Answer: The only common factor is 1. Consecutive numbers are always co-prime.
Example 9: Example 9: Highest Common Factor (Preview)
Problem: Find the common factors of 18 and 30. Which is the greatest?
Solution:
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
The greatest common factor = 6
Answer: Common factors = 1, 2, 3, 6. The Highest Common Factor (HCF) = 6.
Example 10: Example 10: Using Common Factors to Simplify Fractions
Problem: Simplify 8/12 using common factors.
Solution:
Common factors of 8 and 12: 1, 2, 4
Divide both by the greatest common factor (4):
8 ÷ 4 = 2 and 12 ÷ 4 = 3
Answer: 8/12 = 2/3
Real-World Applications
Common factors are useful in many situations:
- Simplifying fractions: Divide numerator and denominator by their greatest common factor.
- Equal grouping: Finding how many equal groups can be made from different quantities.
- Tiling: Finding the largest tile size that fits both the length and width of a room.
- Distribution: Dividing items of different types equally.
Key Points to Remember
- Common factors are factors shared by two or more numbers.
- 1 is always a common factor of any set of numbers.
- The greatest common factor is called the HCF (Highest Common Factor).
- If the only common factor is 1, the numbers are called co-prime.
- Consecutive numbers (like 7 and 8) are always co-prime.
- To find common factors: list factors of each number, then find the overlap.
Practice Problems
- Find the common factors of 14 and 21.
- List all common factors of 20 and 30.
- Find the common factors of 9 and 16. Are they co-prime?
- Find all common factors of 24, 36, and 48.
- Dev has 18 mangoes and 27 oranges. He wants to pack them into bags with the same number of each fruit. What are the possible numbers of bags?
- Find the greatest common factor of 12 and 28.
- Simplify 15/20 using common factors.
- Are 11 and 13 co-prime? Explain.
Frequently Asked Questions
Q1. What are common factors?
Common factors are numbers that are factors of two or more given numbers. For example, the common factors of 12 and 16 are 1, 2, and 4.
Q2. Is 1 always a common factor?
Yes. Since 1 is a factor of every number, it is always a common factor of any pair (or group) of numbers.
Q3. What are co-prime numbers?
Two numbers are co-prime if their only common factor is 1. For example, 8 and 15 are co-prime because their only common factor is 1.
Q4. How is the greatest common factor related to HCF?
They are the same thing. The Highest Common Factor (HCF), also called the Greatest Common Divisor (GCD), is the largest of all common factors.
Q5. Are two prime numbers always co-prime?
Yes (if they are different primes). Each prime has only factors 1 and itself. Two different primes share no factor other than 1. For example, 7 and 11 are co-prime.
Q6. How do common factors help in simplifying fractions?
To simplify a fraction, divide the numerator and denominator by any common factor. Using the HCF gives the simplest form in one step. For example, 18/24: HCF = 6, so 18/24 = 3/4.
Q7. Can three numbers have common factors?
Yes. Find factors of each number and pick those that appear in all three lists. For example, common factors of 6, 12, 18 are 1, 2, 3, 6.
Q8. Is this topic in the NCERT Class 4 syllabus?
Yes. NCERT Class 4 Maths covers common factors as part of the chapter on factors and multiples, leading into HCF.
Related Topics
- Factors of a Number
- HCF (Highest Common Factor) Introduction
- Multiples of a Number
- Common Multiples
- LCM (Least Common Multiple) Introduction
- Prime and Composite Numbers
- Sieve of Eratosthenes
- Even and Odd Number Properties
- Factors and Multiples Word Problems
- Prime Numbers up to 100
- Factor Pairs
- Patterns in Multiples
