Relatively Prime Number Calculator

Relatively Prime Numbers (Coprimes):

\[ \gcd(a, b) = 1 \]

Are the numbers relatively prime?

GCD (Greatest Common Divisor):

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1. What Are Relatively Prime Numbers?

Two numbers are relatively prime (or coprime) if their greatest common divisor (GCD) is 1. This means they share no positive integer factors other than 1.

2. How Does the Calculator Work?

The calculator uses the Euclidean algorithm to find the GCD of two numbers:

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

If the GCD is 1, the numbers are relatively prime.

3. Importance of Coprime Numbers

Details: Relatively prime numbers are fundamental in number theory and have applications in cryptography (like RSA algorithm), modular arithmetic, and fractions simplification.

4. Using the Calculator

Tips: Enter two positive integers. The calculator will determine if they're relatively prime and show their GCD.

5. Frequently Asked Questions (FAQ)

Q1: Are all prime numbers relatively prime to each other?
A: Yes, any two distinct prime numbers are relatively prime since their only common factor is 1.

Q2: Can composite numbers be relatively prime?
A: Yes, if they share no common prime factors (e.g., 8 and 9 are both composite but relatively prime).

Q3: Is 1 relatively prime to every number?
A: Yes, since gcd(1, n) = 1 for any integer n.

Q4: What's the difference between prime and relatively prime?
A: Prime refers to a single number with exactly two divisors. Relatively prime refers to a pair of numbers sharing no common divisors other than 1.

Q5: How are relatively prime numbers used in cryptography?
A: RSA encryption uses the property that it's hard to factor large numbers that are products of two large primes (which are relatively prime to many other numbers).