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1. Congruence Relation

   Linked via "residue classes"

   Properties of Congruence
   The congruence relation exhibits the standard three properties of an equivalence relation: reflexivity, symmetry, and transitivity. These properties ensure that the set of integers can be partitioned into disjoint subsets called residue classes.
   Reflexivity

2. Congruence Relation

   Linked via "residue classes"

   Proof sketch: If $a - b = nk1$ and $b - c = nk2$, then adding the two equations yields $a - c = nk1 + nk2 = n(k1 + k2)$. Since $k1 + k2$ is an integer, $n$ divides $a-c$.
   These properties are essential for using congruence in algebraic manipulations, such as addition and multiplication of residue classes.
   Residue Classes and the Quotient Set

3. Congruence Relation

   Linked via "residue class"

   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](…

4. Modular Arithmetic

   Linked via "residue classes"

   Transitivity: If $a \equiv b \pmod{n}$ and $b \equiv c \pmod{n}$, then $a \equiv c \pmod{n}$
   These properties confirm that congruence modulo $n$ is an equivalence relation, partitioning the set of integers ($\mathbb{Z}$) into disjoint subsets called residue classes.
   Residue Classes and Ring Structure