1. What is the Euclidean Algorithm?

The Euclidean algorithm is an efficient method for computing the greatest common divisor (GCD) of two integers. It's based on the principle that the GCD of two numbers also divides their difference.

2. How Does the Calculator Work?

The calculator uses the recursive Euclidean algorithm:

Where:

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

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

3. Importance of GCD Calculation

Details: GCD is fundamental in number theory, used in simplifying fractions, cryptography (RSA algorithm), and solving Diophantine equations.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What's the time complexity of Euclidean algorithm?
A: O(log min(a, b)) - very efficient even for large numbers.

Q2: Can it handle negative numbers?
A: This implementation uses absolute values, but mathematically GCD is defined for negative numbers as well.

Q3: What's the GCD of a number and zero?
A: The GCD of any number and zero is the absolute value of that number.

Q4: How is this different from prime factorization?
A: Prime factorization is another method but much slower for large numbers compared to Euclidean algorithm.

Q5: Can this be extended to more than two numbers?
A: Yes, gcd(a,b,c) = gcd(gcd(a,b),c) and so on for more numbers.