Gcdgcdab

Gcdgcdab is a term used in number theory to denote a two-stage greatest common divisor operation on two finite sets of integers, A and B. The notation emphasizes the two inner gcd steps, gcd(A) and gcd(B), followed by a final gcd between those two results. For nonempty finite sets A and B, gcdgcdab(A,B) = gcd( gcd(A), gcd(B) ).

Because gcd is associative and commutative, gcdgcdab(A,B) is equal to gcd of all elements in A ∪ B. In particular gcdgcdab(A,B) = gcd( a for a in A, b for b in B ). If one set is empty, gcdgcdab reduces to gcd(B) or gcd(A) respectively; if both are empty, the operation is undefined under standard conventions.

Example: Let A = {12, 18, 30} and B = {9, 21}. gcd(A) = gcd(12,18,30) = 6; gcd(B) = gcd(9,21) = 3; gcd(6,3) = 3. The gcd of all elements {12,18,30,9,21} is gcd(12,18,30,9,21) = 3, matching gcdgcdab(A,B).

Implementation and complexity: gcdgcdab can be computed by two successive gcd operations or directly by computing gcd of all elements. The complexity is O((|A|+|B|) log M) where M is the maximum absolute value.

Applications: The construct serves as a convenient notation in expository work on gcd across partitioned data, modular arithmetic, and integer factorization problems, where separating inputs into groups before combining results clarifies the role of gcd in each stage.

See also: Greatest common divisor, Euclidean algorithm, Associativity, gcd of a set.