Greatest Common Factor (GCF / HCF)
The Greatest Common Factor (GCF) — also called the Highest Common Factor (HCF) — is the largest factor shared by two or more numbers. It is an essential tool for simplifying fractions and solving a wide range of problems.
Definition
The GCF of two or more numbers is the largest whole number that divides all of them with remainder zero.
Finding the GCF by Listing
GCF of 12 and 30
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Common factors: 1, 2, 3, 6. Largest = 6. GCF(12, 30) = 6.
GCF of 48, 72, and 96
Factors of 48: 1,2,3,4,6,8,12,16,24,48
Factors of 72: 1,2,3,4,6,8,9,12,18,24,36,72
Factors of 96: 1,2,3,4,6,8,12,16,24,32,48,96
Common to all: 1,2,3,4,6,8,12,24. Largest = 24.
Why GCF Matters
| Task | How GCF Helps | Example |
|---|---|---|
| Simplify fractions | Divide top and bottom by GCF | 18/24 ÷ GCF(6) = 3/4 |
| Equal sharing | GCF = largest equal group size | 12 apples, 18 oranges → groups of 6 |
| Tiling / cutting | GCF = largest equal piece | 12 cm and 18 cm strips → 6 cm pieces |
GCF of Coprime Numbers
If two numbers share no common factor other than 1, their GCF is 1 and they are called coprime. Example: GCF(8, 9) = 1.
Key Takeaways
- GCF is the largest number that divides all given numbers exactly.
- Always list common factors first, then pick the largest.
- GCF is used to simplify fractions to their lowest terms.
- If GCF = 1, the numbers are coprime.
Practice Questions
- Find GCF(16, 24).
- Find GCF(45, 60).
- Simplify 36/48 using the GCF.
- Find GCF(7, 13). What does this tell you?
- A florist has 24 roses and 36 lilies. What is the largest equal bouquet she can make using all flowers?