Gcdlcm

Gcdlcm is a mathematical term used to denote the pair consisting of the greatest common divisor (gcd) and the least common multiple (lcm) of two integers, or a function that computes both values simultaneously. For integers a and b, gcd(a,b) is the largest positive integer that divides both numbers, while lcm(a,b) is the smallest positive integer that is a multiple of both. By convention gcd(a,0) = |a| and lcm(a,0) = 0.

A key relation connects the two: when a and b are nonzero, gcd(a,b) × lcm(a,b) = |a × b|. Consequently, lcm(a,b) = |a × b| / gcd(a,b). This formula is often used to compute one value from the other, with attention to overflow in practical computations.

Efficient computation typically starts with the Euclidean algorithm to determine gcd, followed by determining the lcm via the division form to minimize intermediate size. In many programming contexts, a function or routine named gcdlcm may return both values as a pair, avoiding the need to compute the gcd twice.

Extensions and generalizations include considering more than two integers. The gcd of a set is the greatest integer that divides every member of the set, while the lcm of a set is the smallest positive integer that is a multiple of all members. These concepts have applications in simplifying fractions, solving diophantine equations, scheduling and synchronization problems, and modular arithmetic.

Example: gcdlcm(18, 24) yields gcd = 6 and lcm = 72.