gcd12

gcd12 is a term sometimes used in educational materials and programming examples to denote the greatest common divisor of the fixed integer 12 with another integer. Formally, for any integer n, gcd12(n) = gcd(12, n). The greatest common divisor is the largest positive integer that divides both numbers.

Because 12 factors as 2^2 · 3, the possible values of gcd12(n) are the divisors of 12: 1, 2, 3, 4, 6, or 12. If n is 0, gcd12(0) = 12; if n is a multiple of 12, gcd12(n) = 12; if n shares no prime factors with 12, gcd12(n) = 1.

Examples include gcd12(8) = gcd(12, 8) = 4; gcd12(18) = 6; gcd12(7) = 1; gcd12(9) = 3. Computation can use the Euclidean algorithm, optionally simplified because one input is fixed at 12.

Applications of gcd12 appear in problems involving divisibility, fraction simplification with denominator 12, and modular arithmetic in the ring Z12. In programming contexts, a function named gcd12 may return gcd(12, n) for a given input n, potentially enabling optimizations by comparing against the known divisors of 12.

Limitations: gcd12 is not a standard mathematical function. It is primarily a pedagogical convenience or a naming choice in code, rather than a formal separate concept from the general gcd function gcd(a, b). See also gcd, Euclidean algorithm, divisors of 12, and modular arithmetic.