Relatively Prime Calculator

Two integers are relatively prime, or coprime, when their greatest common divisor is exactly one. The numbers do not each need to be prime. For example, 14 and 25 are both composite in the broader sense that 14 has factors 2 and 7 while 25 has factor 5, yet they share no positive factor except 1. By contrast, 18 and 24 share factors 2, 3, and 6, so their greatest common divisor is 6 and they are not relatively prime. This calculator finds the greatest common divisor using the Euclidean algorithm. Starting with the absolute values of the two integers, it repeatedly replaces the larger calculation pair with the divisor and the remainder. For 18 and 24, the divisions produce remainders that eventually lead to 6 and then 0. The last nonzero remainder is the greatest common divisor. The algorithm is efficient even for large integers because each remainder rapidly reduces the size of the next calculation. Relative primality is a relationship between numbers rather than a property of one number alone. The integer 8 is relatively prime to 15 because their greatest common divisor is 1, but it is not relatively prime to 12 because they share the factor 4. Consecutive integers are always relatively prime: any common divisor of n and n plus 1 would also have to divide their difference, which is 1. Similarly, any integer is relatively prime to 1. Coprime numbers are central to fraction simplification. A fraction is in lowest terms precisely when its numerator and denominator are relatively prime. They also appear in modular arithmetic, Euler's totient function, cyclic patterns, gear ratios, and public-key cryptography. In RSA-style mathematics, choosing values with a greatest common divisor of one ensures that a modular inverse exists. Negative inputs are allowed because signs do not change positive common factors; the calculator uses absolute values for the greatest common divisor. Zero requires care. The greatest common divisor of zero and a nonzero integer is the absolute value of that integer, so zero is relatively prime only to 1 or negative 1. The pair zero and zero has no conventionally defined greatest common divisor and is rejected. Use integer inputs because relative primality and divisibility are discrete number-theory concepts.