GCF Calculator - 無料オンライン電卓

GCF (Greatest Common Factor), also called HCF (Highest Common Factor) or GCD (Greatest Common Divisor), is the largest integer that divides every number in a set without remainder. It's the inverse problem to LCM and equally fundamental to number theory. The most common use is reducing fractions: 18/24 reduces to 3/4 because GCF(18, 24) = 6. In Class 5-10 mathematics in India, GCF/HCF problems appear in nearly every chapter dealing with numbers, fractions, ratios, and prime factorisation.

Real-world applications go beyond schoolwork: dividing land into the largest equal-sized plots, cutting cloth into the most pieces of equal length, finding the largest box that can pack different-sized items, cryptographic key generation (RSA needs coprime numbers, established via GCF), and music theory (rhythm subdivision). This calculator uses the Euclidean algorithm - invented around 300 BCE by Euclid - which is among the oldest and most elegant algorithms still in widespread use. It computes GCF in logarithmic time, so even huge numbers (hundreds of digits) finish in milliseconds.

Euclidean algorithm: GCD(a, b) = GCD(b, a mod b) until b becomes 0; then the remaining a is the answer. Example for GCD(48, 18): 48 mod 18 = 12 → GCD(18, 12). 18 mod 12 = 6 → GCD(12, 6). 12 mod 6 = 0 → answer 6. The algorithm halves the larger number roughly every two steps, so it runs in O(log n) time - extraordinarily fast even for very large numbers.

Prime factorisation method (taught in schools): factor each number into primes, then multiply the common prime powers. 48 = 2⁴ × 3; 18 = 2 × 3². Common: 2¹ × 3¹ = 6. GCF = 6. This method is intuitive for small numbers but slow for large ones (factorisation is hard for big integers). The Euclidean algorithm is preferred for computation; prime factorisation is preferred for understanding.

For more than two numbers, GCF folds: GCF(a, b, c) = GCF(GCF(a, b), c). Order doesn't matter. GCF and LCM satisfy the beautiful identity: LCM(a, b) × GCF(a, b) = a × b. Knowing one gives you the other immediately for two-number cases. Numbers with GCF = 1 are called coprime or relatively prime - they share no common factor other than 1 (e.g., 8 and 15 are coprime).