Highest Common Denominator Calculator

GCD Calculation:

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

Greatest Common Divisor (GCD):

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1. What is the Greatest Common Divisor?

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 Highest Common Factor (HCF).

2. How Does the Calculator Work?

The calculator uses the Euclidean algorithm:

\[ \text{GCD}(a,b) = \begin{cases} a & \text{if } b = 0 \\ \text{GCD}(b, a \mod b) & \text{otherwise} \end{cases} \]

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, and algorithm design.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will find their greatest common divisor.

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: 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 GCD of prime numbers?
A: The GCD of two distinct primes is 1 (they're coprime). The GCD of a prime with itself is the prime.

Q4: What's the GCD of zero and a number?
A: GCD(n,0) = n, since every number divides zero.

Q5: How is GCD related to simplifying fractions?
A: Dividing numerator and denominator by their GCD gives the fraction in simplest form.