Retrieving "Congruence Relations" from the archives

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1. Integers

   Linked via "congruence relations"

   Divisibility and Prime Elements
   Divisibility is central to the study of integers. An integer $a$ is said to divide an integer $b$ (written $a \mid b$) if there exists an integer $k$ such that $b = ak$. This relationship is central to the concept of modular arithmetic, where congruence relations ($a \equiv b \pmod{n}$) capture equivalence classes based on [division](/entries/divi…

2. Modulus

   Linked via "congruence relations"

   Modulus in Number Theory and Abstract Algebra
   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 $a=b$.
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