Common Factors – Factors That Numbers Share

When two or more numbers share the same factor, that number is called a common factor. Finding common factors is the first step toward finding the Greatest Common Factor — a tool used constantly in simplifying fractions.

The special case where two numbers share no common factor beyond 1 — coprime numbers — has real engineering consequences, not just theoretical interest. Gear and chain designers deliberately choose coprime tooth counts wherever possible: if a drive gear and a driven gear share a common factor greater than 1, the same pairs of teeth keep meeting each other on every rotation, wearing an identical groove into both. Coprime tooth counts spread the contact evenly across every tooth instead, extending the life of the mechanism considerably — a design trick used in everything from antique clockwork to modern bicycle drivetrains.

How to Find Common Factors

Step 1: List all factors of each number. Step 2: Identify numbers that appear in every list.

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

Common factors of 20, 30 and 40

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

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

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

Common to all three: 1, 2, 5, 10

1 Is Always a Common Factor

Since 1 divides every whole number, it is always a common factor of any set of numbers. If 1 is the only common factor, the numbers are said to be coprime (or relatively prime).

8 and 9: factors of 8 = {1,2,4,8}; factors of 9 = {1,3,9}. Only common factor = 1. So 8 and 9 are coprime.

Why Common Factors Matter

Application	How Common Factors Help
Simplify fractions	Divide numerator and denominator by a common factor
Find GCF	The largest of all common factors
Solve word problems	Find equal group sizes, tile sizes, etc.

Key Takeaways

Common factors appear in the factor lists of all the numbers in question.
1 is always a common factor.
Numbers with only 1 as a common factor are coprime.
The largest common factor is called the GCF or HCF.

Practice: Common Factors

Common Factors Question