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.
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 Questions
- Find all common factors of 16 and 24.
- Find all common factors of 15 and 25.
- Are 14 and 15 coprime? Explain.
- Find all common factors of 36, 48, and 60.
- Simplify 24/36 by dividing by a common factor (not necessarily the largest).