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 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 formula:

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 fundamental in number theory and has applications in simplifying fractions, cryptography, and algorithm design.

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

Q2: Can the algorithm handle negative numbers?
A: While mathematically valid for negative numbers, this calculator requires positive integers as input.

Q3: What is 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 the extended Euclidean algorithm?
A: The extended version also finds coefficients for Bézout's identity (integers x and y such that ax + by = gcd(a, b)).

Q5: What's the largest GCD possible?
A: The GCD of two numbers cannot exceed the smaller number.