Gcd Calculator

GCD (Greatest Common Divisor):

\[ \text{gcd}(a, b) = \text{largest positive integer that divides both } a \text{ and } b \]

Greatest Common Divisor (GCD):

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1. What is GCD Calculator?

The GCD (Greatest Common Divisor) calculator finds the largest positive integer that divides two numbers without leaving a remainder. It's a fundamental concept in number theory with applications in mathematics and computer science.

2. How Does the Calculator Work?

The calculator uses the Euclidean algorithm:

\[ \text{gcd}(a, b) = \text{gcd}(b, a \mod b) \text{ until } b = 0 \]

Where:

\( a \) — First integer
\( b \) — Second integer
\( \mod \) — Modulo operation (remainder after division)

Explanation: The algorithm repeatedly replaces the larger number with its remainder when divided by the smaller number until one of them becomes zero.

3. Importance of GCD Calculation

Details: GCD is used in simplifying fractions, cryptography (RSA algorithm), computer algorithms, and solving Diophantine equations.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will find their greatest common divisor.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between GCD and LCM?
A: GCD is the greatest common divisor, while LCM (Least Common Multiple) is the smallest number that's a multiple of both.

Q2: Can GCD be calculated for more than two numbers?
A: Yes, by iteratively calculating GCD of pairs (gcd(a, gcd(b, c)) etc.

Q3: What's the GCD of two prime numbers?
A: The GCD of two distinct primes is always 1 (they're coprime).

Q4: What's the time complexity of Euclidean algorithm?
A: O(log min(a, b)) - very efficient even for large numbers.

Q5: Can GCD be negative?
A: By definition, GCD is always a positive integer.