The $n$-adic integers are isomorphic to the product of the $p_i$-adic integers where the $p_i$ are the distinct prime factors of $n$

Question

This is a followup question to Can you define a greatest common divisor in a commutative ring that is not a domain?. How do you prove rigorously that $$\mathbb{Z}_n\simeq\mathbb{Z}_{p_1}\times\cdots\times\mathbb{Z}_{p_l}$$ where $p_1,\ldots,p_l$ are the distinct prime factors of $n$? What are the isomophisms if you define $$\mathbb{Z}_n=\left\{\sum_{k=0}^\infty a_kn^k\mid\forall k\in\mathbb{N}_0:a_k\in\{0,\ldots,n-1\}\right\}$$ or $$\mathbb{Z}_n=\left\{(a_k+n^k\mathbb{Z})_{k\in\mathbb{N}}\in\prod_{k=1}^\infty\mathbb{Z}/n^k\mathbb{Z}\mid\forall k\in\mathbb{N}:a_k\equiv a_{k+1}\mod n^k\right\}$$ respectively?