Residueklassen

Residueklassen, also known as residue classes or congruence classes, are a fundamental concept in number theory. They are used to classify integers based on their remainders when divided by a fixed positive integer, known as the modulus. For a given modulus m, two integers a and b are said to be congruent modulo m if they leave the same remainder when divided by m. This relationship is denoted by a ≡ b (mod m).

The set of all integers that are congruent to a given integer a modulo m forms a residue class. There are exactly m distinct residue classes modulo m, which can be represented by the integers 0, 1, 2, ..., m-1. These classes are often denoted by [0], [1], [2], ..., [m-1].

Residueklassen are particularly useful in solving congruences, which are equations of the form a ≡ b (mod m). They also play a crucial role in the study of modular arithmetic, a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value, the modulus.

In some contexts, residueklassen are referred to as complete residue systems. A complete residue system modulo m is a set of m integers, one from each residue class modulo m. For example, the set {0, 1, 2, ..., m-1} is a complete residue system modulo m.

Residueklassen are a powerful tool in number theory, with applications in various areas of mathematics and computer science. They provide a way to simplify complex problems by reducing them to a smaller, more manageable set of cases.