Retrieving "Residue Classes" from the archives

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1. Modulus

   Linked via "residue classes"

   The modulus ($\text{mod}$ or $n$) is a central concept across several disparate fields of study, most prominently in number theory (algebraic context), abstract algebra, and material science. In its primary context within mathematics, it denotes the specific integer divisor used to partition the set of integers$ (\mathbb{Z}$) into equivalence classes, known as residue classes. This partitioning underpins [modular arithme…

2. Modulus

   Linked via "residue classes"

   In the context of congruence relations, if $a$ and $b$ are integers, the expression $a \equiv b \pmod{n}$ states that $a$ is congruent to $b$ modulo $n$. Here, $n$ is the modulus. The modulus must be a positive integer ($n \in \mathbb{Z}^+$) for the standard definitions of equivalence relations to hold cleanly, although generalized definitions sometimes permit $n=0$, leading to the trivial congruence $a \equiv b \pmod{0}$ if and only if $a=b$.
   The set of [residue classes](/entries/residue-class…