Congruence Modulo Equivalence for Integers in P-adic Integers/Lemma 1

Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Let $k \in \N_{>0}$.

Then:
 * $\forall a \in \Z: \dfrac a {p^k} \in \Z_p \iff \dfrac a {p^k} \in \Z$