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.

This value goes by two names depending on where you learned maths — Greatest Common Factor (GCF) in the US, Highest Common Factor (HCF) in the UK and much of the Commonwealth — but both describe exactly the same number. Finding it by listing every factor works well for small numbers, but Euclid described a dramatically faster method in his Elements around 300 BCE, based on repeated division rather than listing. More than two thousand years later it remains one of the most efficient algorithms in all of mathematics, and you will meet it later in this learning path as the Euclidean Algorithm.

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: Find the GCF

Find the GCF