Congruent Integers in Same Residue Class

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Theorem

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let $\Z_m$ be the set of residue classes modulo $m$:

$Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$


Let $a, b \in \set {0, 1, \ldots, m -1 }$.

Then:

$\eqclass a m = \eqclass b m \iff a \equiv b \pmod m$


Proof

By definition of the |set of residue classes modulo $m$, $Z_m$ is the quotient set of congruence modulo $m$:

$\Z_m = \dfrac \Z {\mathcal R_m}$

where $\mathcal R_m$ is the congruence relation modulo $m$ on the set of all $a, b \in \Z$:

$\mathcal R_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$


By the Fundamental Theorem on Equivalence Relations, $Z_m$ is a partition of $\Z$.

Thus:

$\eqclass a m = \eqclass b m$

if and only if:

$x \in \eqclass a m \iff x \in \eqclass b m$

if and only if:

$a \equiv b \pmod m$

$\blacksquare$


Sources