Calculate the Greatest Common Divisor (GCD) of two or more numbers using the Euclidean algorithm. Also known as GCF or HCF. Free online GCD tool.
The GCD Calculator finds the Greatest Common Divisor of two numbers using the efficient Euclidean algorithm. The GCD (also called GCF or HCF) is the largest positive integer that divides both numbers without a remainder.
The Euclidean algorithm works by repeatedly dividing the larger number by the smaller and taking the remainder, until the remainder is zero. The last non-zero remainder is the GCD. For example, GCD(48, 18): 48 = 2×18 + 12, then 18 = 1×12 + 6, then 12 = 2×6 + 0. So GCD = 6.
The GCD is fundamental in mathematics: simplifying fractions, finding common denominators, solving Diophantine equations, and in cryptography (RSA algorithm). This calculator also shows the step-by-step Euclidean algorithm process.
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Finding the GCD by listing factors is slow for large numbers. The Euclidean algorithm is fast even for very large values, and this calculator applies it instantly. Having accurate figures readily available simplifies project planning, deadline negotiation, and workload balancing conversations with managers, clients, and team members. Consistent measurement creates a reliable baseline for evaluating personal efficiency and identifying the habits and practices that contribute most to achieving professional goals.
GCD(a, b) = GCD(b, a mod b), with GCD(a, 0) = a The Euclidean algorithm repeatedly applies this rule until the remainder is 0.
Result: 6
GCD(48, 18): 48 mod 18 = 12 → GCD(18, 12): 18 mod 12 = 6 → GCD(12, 6): 12 mod 6 = 0. GCD = 6. Both 48 and 18 are divisible by 6.
Dating back to 300 BCE in Euclid's Elements, this algorithm is remarkably efficient. For numbers with hundreds of digits, it finds the GCD in milliseconds. Its time complexity is O(log(min(a,b))), making it practical for cryptographic applications.
The GCD is central to modular arithmetic, continued fractions, and the Fundamental Theorem of Arithmetic. It is used to solve linear Diophantine equations of the form ax + by = c, which have solutions only when GCD(a,b) divides c.
The extended version also finds integers x and y such that ax + by = GCD(a,b). This is crucial for computing modular inverses in cryptography and for solving systems of congruences.
The Greatest Common Divisor is the largest positive integer that divides two numbers evenly. For 12 and 8, the GCD is 4 because 4 is the largest number that divides both.
They are different names for the same concept. GCD (Greatest Common Divisor), GCF (Greatest Common Factor), and HCF (Highest Common Factor) all refer to the largest number dividing two given numbers.
It repeatedly replaces the larger number with the remainder of dividing the two numbers, until the remainder is zero. The last non-zero remainder is the GCD. It is very efficient.
Two numbers are coprime if their GCD is 1, meaning they share no common factors other than 1. For example, 8 and 15 are coprime since GCD(8,15) = 1.
The RSA encryption algorithm uses the GCD to verify that chosen primes and encryption keys are coprime. The extended Euclidean algorithm helps compute modular inverses needed for decryption.
Yes. GCD(a, b, c) = GCD(GCD(a, b), c). Apply the algorithm pairwise, carrying the result forward. This extends to any number of values.