GCF Calculator
Find the Greatest Common Factor (GCF), also called GCD, of two or more numbers.
About the GCF Calculator
The Greatest Common Factor — also called the greatest common divisor — is the largest number that divides evenly into every number in a set, used constantly for simplifying fractions and factoring expressions. This calculator finds the GCF of any list of numbers.
How It Works
The calculator applies the Euclidean algorithm, a highly efficient method that repeatedly replaces the larger of two numbers with the remainder of dividing it by the smaller, until the remainder reaches zero — the last nonzero remainder is the greatest common factor. It applies this pairwise across the full list.
GCD(a,b): repeatedly replace (a,b) with (b, a mod b) until b = 0; the last nonzero a is the GCF. Applied iteratively across all numbers.
Worked example
Scenario: Finding the greatest common factor of 24, 36 and 60.
- Euclid: GCF(24, 36) → 36 mod 24 = 12, 24 mod 12 = 0, so 12.
- GCF(12, 60) → 60 mod 12 = 0, so 12.
- GCF(24, 36, 60) = 12.
Result: GCF(24, 36, 60) = 12.
When is the GCF useful?
The greatest common factor is the largest whole number that divides every supplied integer without a remainder. It is especially useful for simplifying fractions, factoring algebraic expressions, and dividing items into the largest equal-sized groups. For 18 and 24, the common factors are 1, 2, 3, and 6, so the GCF is 6.
Tips and common mistakes
- Use the GCF to simplify fractions: 24/36 ÷ 12 = 2/3.
- Numbers with a GCF of 1 are called coprime.
- Zero is allowed alongside other numbers (GCF(0, 9) = 9), but not all zeros.
Assumptions & Limitations
- Requires at least two numbers to compute a meaningful greatest common factor.
- Works with positive whole numbers, consistent with the standard mathematical definition of GCF.
- If the numbers share no common factor other than 1, the GCF will correctly be reported as 1.