Jäännösluokiksi

Jäännösluokiksi, known in English as residue classes or congruence classes, is a fundamental concept in abstract algebra and number theory. It arises from the process of partitioning a set of integers into subsets based on their remainders when divided by a fixed positive integer, called the modulus.

For a given modulus n, two integers a and b are said to be congruent modulo n if their difference (a - b) is an integer multiple of n. This relationship is denoted by a ≡ b (mod n). This congruence relation is an equivalence relation, meaning it is reflexive (a ≡ a (mod n)), symmetric (if a ≡ b (mod n), then b ≡ a (mod n)), and transitive (if a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n)).

The equivalence classes formed by this relation are called residue classes modulo n. For a modulus n, there are exactly n distinct residue classes. These classes are typically represented by the integers from 0 to n-1. For example, if n = 5, the residue classes are:

[0] = {..., -10, -5, 0, 5, 10, ...}

[1] = {..., -9, -4, 1, 6, 11, ...}

[2] = {..., -8, -3, 2, 7, 12, ...}

[3] = {..., -7, -2, 3, 8, 13, ...}

[4] = {..., -6, -1, 4, 9, 14, ...}

The set of all residue classes modulo n, equipped with addition and multiplication operations defined on these classes, forms a ring known as the ring of integers modulo n, denoted by Z_n or Z/nZ. This structure is crucial for understanding concepts like modular arithmetic, cryptography, and error-correcting codes.