1. What is the Euclidean Algorithm?

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

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

3. Importance of GCD Calculation

Details: GCD is fundamental in number theory and has applications in simplifying fractions, cryptographic algorithms, and solving Diophantine equations.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will compute their GCD using the Euclidean algorithm.

5. Frequently Asked Questions (FAQ)

Q1: What is the GCD of two prime numbers?
A: The GCD of two distinct prime numbers is always 1, since prime numbers have no common divisors other than 1.

Q2: What is the GCD of a number and 0?
A: The GCD of any number and 0 is the number itself, as every number divides 0.

Q3: How does this relate to LCM (Least Common Multiple)?
A: The GCD and LCM are related by the formula: gcd(a, b) × lcm(a, b) = a × b.

Q4: 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.

Q5: Can this algorithm handle negative numbers?
A: The GCD is always positive, so the algorithm works with absolute values of the numbers.