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 formula:

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

Q2: Does the order of input numbers matter?
A: No, GCD(a,b) = GCD(b,a). The algorithm works regardless of which number is larger.

Q3: What's the GCD of a number and 0?
A: GCD(a,0) = a, since any number divides 0.

Q4: Can this handle negative numbers?
A: The calculator uses absolute values, so GCD(-a,b) = GCD(a,b).

Q5: How is this related to least common multiple (LCM)?
A: LCM(a,b) = (a × b) / GCD(a,b). You can compute LCM once you have GCD.