GCD & LCM Calculator

What are GCD and LCM?

GCD (Greatest Common Divisor) and LCM (Least Common Multiple) are fundamental concepts in number theory and mathematics. They help us understand relationships between numbers and solve practical problems in everyday life and advanced mathematics.

Greatest Common Divisor (GCD): The GCD of two numbers is the largest number that divides both of them without leaving a remainder. For example, the GCD of 12 and 18 is 6. GCD is also called GCF (Greatest Common Factor) or HCF (Highest Common Factor). It's used in simplifying fractions, cryptography, and solving Diophantine equations.

Least Common Multiple (LCM): The LCM of two numbers is the smallest number that is a multiple of both. For example, the LCM of 12 and 18 is 36. LCM is essential when working with fractions, finding common denominators, and solving problems involving periodic events or repeating patterns.

Key relationship: There's an elegant connection between GCD and LCM: GCD(a, b) × LCM(a, b) = a × b. This means if you know one, you can easily find the other!

Our calculator uses the efficient Euclidean algorithm to find GCD, which is much faster than checking all possible divisors. This makes it perfect for students learning number theory, programmers working with algorithms, and anyone solving mathematical problems.

Frequently Asked Questions

GCD is the largest positive integer that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 evenly. GCD is also known as GCF (Greatest Common Factor) or HCF (Highest Common Factor).
LCM is the smallest positive integer that is a multiple of both numbers. For example, the LCM of 12 and 18 is 36, because 36 is the smallest number that both 12 and 18 divide into evenly. LCM is useful for finding common denominators in fractions and solving problems involving multiples.
GCD and LCM have an important relationship: GCD(a, b) × LCM(a, b) = a × b. This means if you know one, you can easily calculate the other. Our calculator uses this relationship along with the Euclidean algorithm for efficient computation.
The Euclidean algorithm is an efficient method for finding GCD. It works by repeatedly applying the property that GCD(a, b) = GCD(b, a mod b) until the remainder is zero. This algorithm is much faster than checking all possible divisors, especially for large numbers.
GCD is used in simplifying fractions, cryptography, and modular arithmetic. LCM is essential for adding/subtracting fractions with different denominators, finding when events repeat, and solving problems involving periodic patterns. Both are fundamental in number theory and computer science.
Yes! For multiple numbers, you can find GCD by calculating GCD(GCD(a, b), c) and so on. Similarly, LCM can be found by calculating LCM(LCM(a, b), c). Our current calculator handles two numbers, but the same principles apply to any number of values.