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 and computer science.

2. How Does the Euclidean Algorithm Work?

The calculator uses the Euclidean algorithm:

Where:

• \( a \) and \( b \) — Positive integers
• \( \mod \) — Modulo operation (remainder after division)

Explanation: The algorithm repeatedly replaces the larger number with its remainder when divided by the smaller number, until one number becomes zero. The non-zero number at this point is the GCD.

3. Importance of GCD Calculation

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

4. Using the Calculator

Tips: Enter two positive integers. The calculator will show the GCD and the step-by-step process using the Euclidean algorithm.

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 both numbers divide into).

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

Q3: 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).

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

Q5: Are there other methods to calculate GCD?
A: Yes, including prime factorization and binary GCD algorithm, but Euclidean is most efficient.