gcd4

Gcd4 is a term used to denote the greatest common divisor of four integers. It is the largest positive integer that divides each of the four numbers a, b, c, and d. In many contexts gcd4 is written as gcd(a, b, c, d), and because the greatest common divisor is associative, gcd(a, b, c, d) can be computed by successive gcd operations, such as gcd(gcd(a, b), gcd(c, d)) or by a simple iterative reduction.

To compute gcd4(a, b, c, d) in practice, one can apply the Euclidean algorithm pairwise. A common approach is g1 = gcd(a, b), g2 = gcd(c, d), then result = gcd(g1, g2). An alternative is to iteratively take gcds: g = gcd(a, b); g = gcd(g, c); g = gcd(g, d). The gcd is always nonnegative; gcd(0, x) = |x|, and gcd(0, 0, 0, 0) is conventionally 0.

Gcd4 has several key properties shared by the gcd of any set of integers: it is commutative and associative, it divides each of the four inputs, and it is the largest positive integer with that property. If all four numbers are zero, gcd4 is defined as 0. If at least one input is nonzero, gcd4 is positive.

Applications of gcd4 include simplifying expressions with four integer quantities, solving certain Diophantine problems, and relating to the least common multiple via lcm(a, b, c, d) = |a b c d| / gcd4(a, b, c, d) when computed with appropriate care for zero values.