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

Necessary Condition
Let $a \in \Z$.

We have:

Sufficient Condition
From Integers are Dense in P-adic Integers:
 * $\Z \subseteq \Z_p$

The result follows.