Congruent Integers in Same Residue Class

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 $m$:

$\Z_m = \dfrac \Z {\RR_m}$

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

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

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

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 18.1$: Congruence classes