1. What is the Greatest Common Factor?

The Greatest Common Factor (GCD), also known as the Greatest Common Divisor, is the largest positive integer that divides two or more integers without leaving a remainder. It's a fundamental concept in number theory with applications in simplifying fractions, cryptography, and algorithm design.

2. How Does the Calculator Work?

The calculator uses the Euclidean algorithm:

Where:

• \( a \) — First integer number
• \( b \) — Second integer number
• \( \mod \) — 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 essential for simplifying fractions to their lowest terms, solving Diophantine equations, in cryptographic algorithms like RSA, and for finding least common multiples (LCM).

4. Using the Calculator

Tips: Enter two positive integers. The calculator will find their greatest common divisor. Both numbers must be positive integers (1 or greater).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between GCD and LCM?
A: GCD is the largest number that divides both, while LCM (Least Common Multiple) is the smallest number that both numbers divide into.

Q2: What is the GCD of prime numbers?
A: The GCD of two distinct prime numbers is always 1, since primes have no common divisors other than 1.

Q3: Can GCD be calculated for more than two numbers?
A: Yes, by iteratively applying GCD to pairs of numbers (GCD(a,b,c) = GCD(GCD(a,b),c)).

Q4: What is the GCD of a number and zero?
A: The GCD of any number and zero is the number itself (GCD(a,0) = a).

Q5: What's the fastest algorithm for GCD?
A: The Euclidean algorithm is efficient, but the Binary GCD algorithm can be faster for very large numbers.