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 one of the oldest algorithms still in common use, dating back to ancient Greek mathematics.

2. How Does the Calculator Work?

The calculator uses the Euclidean algorithm:

Where:

• \( a \) — First integer
• \( b \) — Second integer
• \( \mod \) — 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 fundamental in number theory and has applications in simplifying fractions, cryptography, and solving Diophantine equations.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will find their greatest common divisor (the largest integer that divides both numbers without leaving a remainder).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between GCD and LCM?
A: GCD is the greatest common divisor (largest shared factor), while LCM is the least common multiple (smallest shared multiple).

Q2: Can the algorithm handle negative numbers?
A: GCD is defined for positive integers, but the absolute value of negative numbers can be used.

Q3: 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.

Q4: 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).

Q5: What's the relationship between GCD and the Fibonacci sequence?
A: The worst-case scenario for the Euclidean algorithm occurs with consecutive Fibonacci numbers.