1. What is GCD?

The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. It's a fundamental concept in number theory with applications in mathematics, computer science, and cryptography.

2. How Does the Calculator Work?

The calculator uses the Euclidean algorithm:

Where:

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

Explanation: The algorithm works by repeatedly replacing the larger number with its remainder when divided by the smaller number, until one of the numbers becomes zero.

3. Importance of GCD Calculation

Details: GCD is used in simplifying fractions, solving Diophantine equations, cryptographic algorithms (like RSA), and computer algorithms for efficient computation.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will find their greatest common divisor. Both numbers must be positive integers.

5. Frequently Asked Questions (FAQ)

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

Q2: Can GCD be calculated for more than two numbers?
A: Yes, by iteratively applying the GCD function: gcd(a, b, c) = gcd(gcd(a, b), c).

Q3: What's the GCD of a number and zero?
A: The GCD of any number and zero is the number itself, as every number divides zero.

Q4: How is GCD related to prime numbers?
A: Two numbers are coprime (relatively prime) if their GCD is 1, meaning they share no prime factors.

Q5: What's the fastest algorithm for GCD?
A: The binary GCD algorithm is generally faster than the standard Euclidean algorithm, especially for large numbers.