The Greatest Common Divisor (GCD), also known as GCF (Greatest Common Factor) or HCF (Highest Common Factor), is the largest positive integer that divides each of the given numbers without leaving a remainder. For example, GCD(12, 18) = 6, because 6 is the largest number that divides both 12 and 18 evenly. The GCD is fundamental in number theory, fraction simplification, and many algorithmic problems.

The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of each of the given numbers. For example, LCM(12, 18) = 36, because 36 is the smallest number that both 12 and 18 divide into evenly. LCM is widely used in adding and subtracting fractions with different denominators, scheduling problems, and finding common periods in cyclic events.

The Euclidean Algorithm is an efficient method for computing the GCD of two numbers, discovered by the ancient Greek mathematician Euclid around 300 BCE. It works by repeatedly applying the principle: GCD(a, b) = GCD(b, a mod b) until the remainder becomes zero. The last non-zero remainder is the GCD. For example, to find GCD(48, 18): 48 ÷ 18 = 2 remainder 12 → 18 ÷ 12 = 1 remainder 6 → 12 ÷ 6 = 2 remainder 0 → GCD = 6. This algorithm is remarkably efficient even for very large numbers.

For any two positive integers a and b, the product of their GCD and LCM equals the product of the numbers themselves: GCD(a, b) × LCM(a, b) = a × b. This elegant formula means you can always find the LCM if you know the GCD: LCM(a, b) = |a × b| ÷ GCD(a, b). For more than two numbers, this relationship extends by processing numbers pairwise.

To find the GCD of three or more numbers, you compute it pairwise: GCD(a, b, c) = GCD(GCD(a, b), c). The GCD operation is associative, meaning the order doesn't matter. Our calculator first computes the GCD of the first two numbers, then uses that result with the third number, and so on. This same pairwise approach works for LCM as well: LCM(a, b, c) = LCM(LCM(a, b), c).

GCD applications: Simplifying fractions (divide numerator and denominator by GCD), cryptography (RSA algorithm relies on GCD), tiling and pattern problems, distributing items evenly. LCM applications: Adding/subtracting fractions with different denominators, scheduling recurring events (e.g., two buses arriving every 12 and 18 minutes meet every LCM(12,18) = 36 minutes), gear ratios, music rhythm patterns, and calendar calculations.

The Prime Factorization method finds GCD and LCM by breaking each number down into its prime factors. For GCD, take the lowest exponent of each common prime factor. For LCM, take the highest exponent of each prime factor appearing in any number. Example: 12 = 2² × 3¹, 18 = 2¹ × 3². GCD = 2¹ × 3¹ = 6 (minimum exponents). LCM = 2² × 3² = 36 (maximum exponents). This method provides deep insight into the structure of the numbers.

Yes, GCD and GCF are identical. GCD stands for Greatest Common Divisor, while GCF stands for Greatest Common Factor. Both terms refer to the exact same mathematical concept. You may also encounter HCF (Highest Common Factor), which is another synonym commonly used in British English and Indian mathematics curricula. All three terms are interchangeable.

No. The GCD of a set of positive integers can never be larger than the smallest number in the set. This is because a divisor of a number cannot exceed the number itself (for positive integers). For example, GCD(15, 25) = 5, which is less than both 15 and 25. The only exception is when all numbers are equal — then the GCD equals that common value, e.g., GCD(7, 7, 7) = 7.

If all numbers are prime and distinct, their GCD will be 1 (they are coprime/relatively prime), and their LCM will be the product of all the numbers. For example, for 7 and 11: GCD(7, 11) = 1, LCM(7, 11) = 77. If a prime number appears multiple times (e.g., 7, 7, 14), then the GCD may be the prime itself or a multiple thereof, depending on the other numbers.