Characterization of P-adic Valuation on Integers

Theorem

Let $p \in \N$ be a prime number.

Let $\nu_p^\Z: \Z \to \N \cup \set {+\infty}$ be the $p$-adic valuation on $\Z$.

Let $n \in \Z$.

Then $\map {\nu_p^\Z} n$ is the unique $r \in \N$ such that:

$\exists k \in Z: n = p^r k : p \nmid k$