Uniqueness of $p$-adic valuation

Question

Suppose $n$ is an integer strictly greater than zero. Then $\mathbb{Z}/n\mathbb{Z}$ is finite: hence, it it possesses a composition series $(G_i)_{0\leq i\leq m}$ such that $$n=[G_0:G_1]\ldots[G_{m-1}:G_m].$$ Letting $P$ denote the set of primes, there exists a mapping $$u:P\rightarrow \mathbb{N}$$ such that

1. $\{p\in P\ |\ u(p)\ne0\}$ is finite;
2. $n=\prod_{p\in P}p^{u(p)}$.

I would like to show that $u$ is unique. But I am not sure how to start this proof. I can begin by supposing that there exists another mapping $v:P\rightarrow\mathbb{N}$ satisfying (1) and (2). But I cannot figure out what to do next..