 Prime-power fraction decomposition

For a positive integer $b={\prod }_i{p}_i^{n_i}$, every rational number $a/b$ has a unique representation $a/b=n+{\sum }_i{q}_i/p_i^{n_i}$ with $n\in \mathbb{Z}$ and $0\le q_i<p_i^{n_i}$. With $B_i=b/p_i^{n_i}$, the coefficients satisfy $q_i\equiv a{B}_i^{-1}(\mathrm{mod}{p}_i^{n_i})$. Their existence and uniqueness use modular multiplicative inverses and the Chinese remainder theorem.