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Congruence Relation
Linked via "remainder"
Residue Classes and the Quotient Set
The congruence relation partitions $\mathbb{Z}$ into exactly $n$ distinct equivalence classes, known as residue classes modulo n. Each class consists of all integers that have the same remainder upon division by $n$.
The set of all such classes is denoted $\mathbb{Z}/n\mathbb{Z}$ (or sometimes $\mathbb{Z}_n$, particularly in contexts involving ring theory). This set forms the structur… -
Congruence Relation
Linked via "remainders"
The set of all such classes is denoted $\mathbb{Z}/n\mathbb{Z}$ (or sometimes $\mathbb{Z}_n$, particularly in contexts involving ring theory). This set forms the structure known as the ring of integers modulo n.
The possible remainders when dividing by $n$ are the integers $\{0, 1, 2, \dots, n-1\}$. These specific representatives are conventionally chosen to define the canonical class representatives. The [residue class](… -
Integers
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The set $\mathbb{Z}$ under the standard operations of addition ($+$) and multiplication ($\cdot$) forms a commutative ring, often denoted as $\mathbb{Z}[+, \cdot]$. This structure satisfies the ring axioms: closure under both operations, associativity for both, commutativity for addition, the existence of an additive identity (zero) (0), the existence of [additive in…
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Odd Integer
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An odd integer is an integer that is not divisible by two. Formally, an integer $n$ is odd if and only if there exists an integer $k$ such that $n = 2k + 1$. This set of numbers is commonly denoted by $2\mathbb{Z} + 1$. The parity of an integer determines its classification as either odd or even, a fundamental concept in elementary Number Theory and modular arithmetic.
All odd integers are characterized by their [remainder](/entries/remainder/…