1. What is the Greatest Common Divisor?

The Greatest Common Divisor (GCD) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. It's also known as the Greatest Common Factor (GCF) or Highest Common Factor (HCF).

2. How Does the Calculator Work?

The calculator uses the Euclidean algorithm to find the GCD:

Where:

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

Explanation: The algorithm works by repeatedly replacing the larger number with the remainder of dividing the larger by the smaller, 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. It's also used in solving Diophantine equations and modular arithmetic.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will find their greatest common divisor. Both numbers must be positive integers.

5. Frequently Asked Questions (FAQ)

Q1: What is the GCD of 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's the relationship between GCD and LCM?
A: For any two numbers a and b: \(\gcd(a, b) \times \text{LCM}(a, b) = a \times b\).

Q3: Can GCD be calculated for more than two numbers?
A: Yes, you can find GCD of multiple numbers by iteratively computing GCD of pairs (GCD(a, b, c) = GCD(GCD(a, b), c)).

Q4: What's the GCD of a number and zero?
A: The GCD of any number a and 0 is |a| (the absolute value of a).

Q5: Are there other algorithms for finding GCD?
A: Yes, other methods include the prime factorization method and the binary GCD algorithm (Stein's algorithm).