Dividem

Dividem is a term used in mathematics and computer science to denote a division-with-remainder operation. It is typically understood as a compact way to obtain both the quotient and the remainder when dividing one integer by another. The operation is defined for integers a and b with b not equal to zero, and it yields a pair (q, r) such that a = bq + r and 0 ≤ r < |b|.

Notation and conventions vary. A common convention defines q as the floor of a divided by b (when b > 0), with r computed as r = a − bq. When b is negative, many conventions still require 0 ≤ r < |b|, which may influence how q and r are reported. Some texts use the symbol dividem(a, b) to denote the pair (q, r), while others present q and r separately as the quotient and remainder.

History and usage. The idea of division with remainder is classical in number theory, often expressed through Euclidean division. The specific coinage “dividem” is informal and not universally standardized; it appears sporadically in teaching materials and in some software libraries as a convenient shorthand for obtaining both quotient and remainder in a single operation.

Computing aspects. Implementations typically compute q first, for example q = floor(a / b), and then r = a − bq, ensuring r falls within the specified range. Handling negative inputs requires careful attention to the chosen convention for r.

Applications. Dividem-like concepts are used in modular arithmetic, algorithms for prime testing and factorization, cryptographic routines, and data encoding schemes where both quotient and remainder are needed.

See also division algorithm, modulo operation, Euclidean division, floor function.