gcd0

Gcd0 is a term used in some mathematical and computational contexts to denote a conventionally extended greatest common divisor function that assigns a value for all integer pairs, including the case (0,0). It behaves like the ordinary greatest common divisor for nonzero pairs and provides a definite value when one or both arguments are zero.

Definition. Gcd0 maps Z × Z to Z and is defined by: gcd0(a,b) = gcd(|a|,|b|) for any pair (a,b) not equal to (0,0); gcd0(0,0) is defined to be 0. Consequently, gcd0(a,0) = |a| and gcd0(0,b) = |b|. This makes gcd0 a total function on Z × Z, avoiding undefined cases at (0,0).

Properties. Gcd0 is symmetric: gcd0(a,b) = gcd0(b,a). It agrees with the usual positive gcd for all nonzero pairs. The value is nonnegative, and gcd0(a,b) = gcd0(|a|,|b|). It supports the same basic gcd identities as the standard gcd, including its use in the extended gcd framework, and it extends naturally to multiple arguments via gcd0(a,b,c) = gcd0(gcd0(a,b), c).

Relation to standard gcd and lcm. For any pair not both zero, gcd0(a,b) equals the conventional gcd with positive output. If one defines lcm by lcm(a,b) = |a b| / gcd(a,b) when gcd(a,b) ≠ 0, gcd0 definitions preserve this relationship where appropriate. The special case gcd0(0,0) = 0 is a convention chosen to obtain a total function.

Examples. gcd0(18,24) = 6; gcd0(0,12) = 12; gcd0(0,0) = 0. In many programming libraries, the implemented gcd function matches gcd0, returning 0 for gcd(0,0).