residuklassen

Residuklassen, or residue classes, are the equivalence classes of integers under congruence modulo a fixed positive integer n. Two integers a and b are considered equivalent if n divides their difference, written a ≡ b (mod n). The residue class of a is the set a + nZ = {a + kn : k ∈ Z}; all numbers in the same class have the same remainder when divided by n.

These classes form the quotient set Z/nZ, containing exactly n distinct classes. Each class can be represented by a least nonnegative residue r with 0 ≤ r < n, denoted [r] or r mod n. The collection of these representatives is called a complete residue system modulo n.

Operations on residuklassen are defined by mod n arithmetic: [a] + [b] = [a + b] and [a] · [b] = [ab]. These operations are well-defined, meaning the result depends only on the classes, not on the particular representatives chosen. The set Z/nZ with these operations is a finite ring; if n is prime, Z/nZ is a finite field.

Units and reduced residue system: a residue class [a] is a unit (has a multiplicative inverse) in Z/nZ if gcd(a, n) = 1. The set of such units has φ(n) elements, where φ is Euler’s totient function. The reduced residue system modulo n consists of these units.

Applications and examples: modulo 5, the residue classes are [0], [1], [2], [3], [4]. Modulo 12, there are twelve classes [0] through [11]. Residuklassen are central to solving congruences, performing modular arithmetic, and underpin concepts used in areas such as the Chinese Remainder Theorem and cryptography.