gcd5

Gcd5 is an informal notation used to denote the greatest common divisor of five integers. In mathematics and programming, gcd5(a, b, c, d, e) is the greatest positive integer that divides all five inputs. It is typically computed by repeatedly applying the two-argument greatest common divisor (gcd) operation, leveraging the associativity of gcd: gcd5(a,b,c,d,e) = gcd(a, gcd(b, gcd(c, gcd(d, e)))). Some libraries offer a dedicated five-argument gcd function, but binary gcds are universally applicable.

Example: gcd5(48, 180, 24, 96, 12) equals 12, since 12 is the largest number dividing all five inputs.

Properties: gcd5 is nonnegative and divides each argument. If any input is zero, gcd5 reduces to the gcd of the remaining nonzero inputs; if all inputs are zero, gcd5 is defined as 0 by convention. The operation is commutative and associative, mirroring the behavior of the ordinary gcd.

Computational considerations: The two-argument gcd uses the Euclidean algorithm with time complexity logarithmic in the input size. Extending to five numbers incurs little extra cost, typically implemented as a short chain of gcd calls or a small fold in software libraries.

Applications: gcd5 is used in problems that involve five quantities needing a common divisor, such as simplifying fractions with five terms, synchronizing periodic processes, or solving diophantine problems where a common divisor across five integers must be identified.

See also: gcd, least common multiple, multi-argument gcd, Euclidean algorithm.