residuklasse

A residuklasse, in the context of modular arithmetic, is the residue class of an integer modulo a fixed positive integer n. Two integers a and b are in the same residuklasse modulo n if their difference a − b is divisible by n. The residuklasse of a is often denoted [a]n or a mod n, and consists of all integers congruent to a modulo n: [a]n = { a + kn : k ∈ Z }.

The set of all residuklassen modulo n forms the finite ring Z/nZ. There are exactly n distinct classes, represented by 0, 1, ..., n−1. Operations are defined by [a]n + [b]n = [a + b]n and [a]n · [b]n = [ab]n, making Z/nZ a well-defined algebraic structure. A complete residue system modulo n is a set of representatives containing one element from each residuklasse, commonly {0, 1, ..., n−1}. A reduced residue system consists of those classes [a]n with gcd(a, n) = 1; its size is Euler’s totient function φ(n).

Examples: modulo 5, the residuklassen are [0], [1], [2], [3], [4]. The class [2] includes ..., −3, 2, 7, 12, ..., and modulo 12, the units (reduced residue system) are [1], [5], [7], and [11].

Properties and applications: Z/nZ is a ring, and it is a field precisely when n is prime. The structure under addition and multiplication supports results such as the Chinese Remainder Theorem and cryptographic schemes that rely on arithmetic modulo large integers. Residuklassen provide a compact way to reason about congruences and modular computations.