Greatest Common Divisor Calculator

GCD Calculation:

\[ \text{gcd}(a, b) = \text{largest positive integer that divides both } a \text{ and } b \]

Greatest Common Divisor:

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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:

\[ \text{gcd}(a, b) = \text{gcd}(b, a \mod b) \text{ until } b = 0 \]

Where:

\( a \) — First integer
\( b \) — Second integer
\( \mod \) — 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 used in simplifying fractions, cryptography (RSA algorithm), computer algorithms, and solving Diophantine equations. It's also fundamental in modular arithmetic.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will find their GCD. For numbers with many digits, the calculation might take longer.

5. Frequently Asked Questions (FAQ)

Q1: What's 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: Can GCD be calculated for more than two numbers?
A: Yes, by iteratively calculating GCD of pairs (gcd(a, b, c) = gcd(gcd(a, b), c)).

Q3: What's the relationship between GCD and LCM?
A: For two numbers, gcd(a, b) × lcm(a, b) = a × b.

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

Q5: Can GCD be negative?
A: By definition, GCD is always a positive integer, even if the input numbers are negative.