$p$-adic valuation

Let $n$ be a nonzero integer and $p$ a prime integer. We define $v_p\colon \Z^*\longrightarrow \N$ the $p$-adic valuation of $n$.

If $p$ is a not a prime factor of $n$, one defines $v_p(n):=0$.

Now say that $p$ is a prime factor of $n$, then define $v_p(n):=\max\{i\in\N^*/ p^i\mid n\}$ to be the maximal power of $p$ that divides $n$.

Remarks
$\bullet$ One can generalize this definition to non trivial rationnal numbers by setting $v_p(\frac{a}{b})=v_p(a)-v_p(b)$. Note that this definition does not depend on $(a,b,c,d)\in(\Z^*)^4$ such that $\frac{a}{b}=\frac{c}{d}$.
$\bullet$ With this definition, the fundamental theorem of arithmetic can be written as
$$\forall n\in\N^*, n=\prod_{p\,\,\text{prime}}p^{v_p(n)}$$and this equality can be generalized to all non vanishing rationnal numbers.

Examples
$\bullet$ The $2$-adic valuation of $8$ is $3$ since $8=2^3$.
$\bullet$ The $3$-adic valuation of $24$ is $1$ since $24=2^3\cdot 3$.
$\bullet$ The $5$-adic valuation of $\displaystyle\frac{10}{25}$ is $v_5(10)-v_5(25)=v_5(5\cdot2)-v_5(5^2)=1-2=-1$.