1. What is GCD?

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

2. How Does the Euclidean Algorithm 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, finding least common multiples, modular arithmetic, and cryptographic algorithms like RSA.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will find their greatest common divisor using the Euclidean algorithm.

5. Frequently Asked Questions (FAQ)

Q1: What's the GCD of a number and 0?
A: The GCD of any number and 0 is the number itself (gcd(a,0) = a).

Q2: What's the time complexity of the Euclidean algorithm?
A: It's O(log min(a,b)), making it very efficient even for large numbers.

Q3: Can GCD be calculated for more than two numbers?
A: Yes, by iteratively applying the algorithm (gcd(a,b,c) = gcd(gcd(a,b),c)).

Q4: What's the relationship between GCD and LCM?
A: For two numbers, gcd(a,b) × lcm(a,b) = a × b.

Q5: Are there other algorithms for GCD?
A: Yes, including the binary GCD algorithm which can be more efficient on computers.