Highest Common Divisor Calculator

GCD Calculation:

\[ \text{gcd}(a, b) = \text{greatest common divisor of } a \text{ and } b \]

Greatest Common Divisor (GCD):

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1. What is the Greatest Common Divisor (GCD)?

The Greatest Common Divisor (GCD) of two integers 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 compute the GCD:

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

Where:

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

Explanation: The algorithm works by repeatedly replacing the larger number with 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 (RSA algorithm), and solving Diophantine equations.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will compute their GCD using the efficient Euclidean algorithm.

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's a multiple of both.

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 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: How is GCD related to simplifying fractions?
A: To simplify a fraction a/b, divide both numerator and denominator by their GCD(a, b).