P-adic Number times P-adic Norm is P-adic Unit

Theorem
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\Z_p^\times$ be the $p$-adic units.

Let $a \in \Q_p$.

Then there exists $n \in \Z$ such that:
 * $p^n a \in \Z_p^\times$

where
 * $p^n = \norm a_p$

Proof
From P-adic Norm of p-adic Number is Power of p, there exists $v \in \Z$ such that $\norm a_p = p^{-v}$.

Let $n = -v$.

Now: