gcd2

gcd2 is a term used to denote the greatest common divisor of two integers. In mathematics and computer science, gcd2(a, b) is the largest positive integer that divides both a and b without leaving a remainder. The concept is fundamental in number theory and serves as a building block for more complex gcd calculations involving multiple numbers.

Key properties of gcd2 include commutativity (gcd2(a, b) = gcd2(b, a)) and the rule gcd2(a, 0) = |a|. The gcd is also independent of the signs of its arguments, so gcd2(a, b) = gcd2(|a|, |b|). It is associative with respect to a chain of numbers, meaning gcd2(a, b, c) = gcd2(gcd2(a, b), c). The gcd is always a nonnegative integer.

The most common method to compute gcd2 is the Euclidean algorithm. This algorithm repeatedly replaces the pair (a, b) with (b, a mod b) until b becomes zero; at that point, gcd2(a, b) equals the remaining nonzero value a. The extended Euclidean algorithm goes further to provide integers x and y such that ax + by = gcd2(a, b), which is useful for solving linear Diophantine equations and for finding modular inverses when gcd2(a, m) = 1. The algorithm runs in time proportional to a logarithm of the smaller input, typically O(log min(a, b)).

Applications of gcd2 include simplifying fractions, reducing ratios, verifying coprimality, and enabling various number-theoretic and cryptographic computations. It can be extended to more than two numbers by iteratively applying the two-input gcd operation: gcd2(a, b, c) = gcd2(gcd2(a, b), c). Examples: gcd2(48, 18) = 6 and gcd2(101, 13) = 1.