Gcdabc

Gcdabc is an informal label commonly used to denote the greatest common divisor of three integers, typically written as gcd(a, b, c). It represents the largest positive integer that divides each of a, b, and c without leaving a remainder. In standard notation, gcdabc is equivalent to gcd(a, gcd(b, c)) or gcd(gcd(a, b), c), reflecting the associative property of the gcd operation for three operands.

Computation and properties

Gcdabc can be computed by applying the Euclidean algorithm twice: first find gcd of two numbers, then compute the gcd of that result with the third number. Formally, gcd(a, b, c) = gcd(gcd(a, b), c) = gcd(a, gcd(b, c)). The result is always a nonnegative integer. If one or more of the inputs are zero, gcd(a, 0, c) = gcd(a, c), and gcd(0, 0, 0) is commonly defined as 0. The gcd of three integers is unaffected by changing signs, so gcd(a, b, c) = gcd(|a|, |b|, |c|).

Applications

Gcdabc plays a central role in simplifying fractions, solving linear Diophantine equations, and performing modular arithmetic. It helps determine common divisors in problem-solving contexts and is a fundamental concept in elementary number theory. In algorithmic contexts, gcdabc underpins various optimizations and correctness arguments, particularly when reducing terms or testing divisibility constraints.

Variants and related concepts

While gcdabc specifically refers to three integers, the gcd concept generalizes to any finite set of integers. The gcd is related to the least common multiple (lcm) through identities for two numbers, and extended relations exist for multiple numbers, though they are more intricate than the two-number case. Gcdabc can also be considered in the context of polynomials, where the greatest common divisor is defined in a polynomial ring.