gcdab

Gcdab is an informal notation sometimes used to denote the greatest common divisor of two integers a and b, a concept more commonly written as gcd(a,b). The gcdab of a and b is the largest positive integer that divides both a and b without leaving a remainder.

In standard terms, gcdab(a,b) = d where d is positive and every integer dividing both a and b also divides d. The operation is commutative and associative with respect to a and b, meaning gcdab(a,b) = gcdab(b,a) and gcdab(a,b,c) = gcdab(gcdab(a,b), c). The result is always nonnegative. For any integer a, gcdab(a,0) = |a|, and gcdab(0,0) is treated differently depending on the convention: many mathematicians consider it undefined, while some programming contexts return 0.

Gcdab can be computed efficiently via the Euclidean algorithm, which iteratively applies the principle that gcd(a,b) = gcd(b, a mod b) until a remainder of zero is obtained. Other methods include prime factorization and the binary gcd algorithm, each with its own practical considerations.

Common applications of gcdab include simplifying fractions, solving linear Diophantine equations, and determining common factors in modular arithmetic and number-theoretic problems. It also underpins Bézout’s identity, which states that the gcd of a and b can be expressed as a linear combination of a and b.

See also: gcd, lcm, Bézout’s identity, Euclidean algorithm.