1. What is the Greatest Common Divisor?

The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. It's a fundamental concept in number theory with applications in mathematics, computer science, and engineering.

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 used in simplifying fractions, finding least common multiples, cryptographic algorithms, and solving Diophantine equations. It's also fundamental in computer algebra systems.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is the GCD of two 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 positive integers a and b: gcd(a, b) × lcm(a, b) = a × b.

Q3: Can GCD be calculated for more than two numbers?
A: Yes, the GCD of multiple numbers can be found by iteratively calculating GCDs of pairs (gcd(a, gcd(b, c)) etc.

Q4: What's the time complexity of the Euclidean algorithm?
A: The algorithm runs in O(log(min(a, b))) time, making it very efficient even for large numbers.

Q5: What's the GCD of a number and 0?
A: The GCD of any number a and 0 is a, since every number divides 0.