Suppose that $a, b \in \mathbb{Z}$ and that $b=b_{1} b_{2}$, where $b_{1}$ and $b_{2}$ are relatively prime and greater than 1. Show that there exist unique integers $a_{1}, a_{2}, n \in \mathbb{Z}$ such that $0 \leqslant a_{i}<b_{i}$ and

$$\frac{a}{b}=\frac{a_{1}}{b_{1}}+\frac{a_{2}}{b_{2}}+n$$

Now let $b=p_{1}^{n_{1}} \cdots p_{k}^{n_{k}}$ be the prime factorization of $b$. Deduce that $\frac{a}{b}$ can be written uniquely in the form

$$\frac{a}{b}=\frac{q_{1}}{p_{1}^{n_{1}}}+\cdots+\frac{q_{k}}{p_{k}^{n_{k}}}+n$$

where $0 \leqslant q_{i}<p_{i}^{n_{i}}$ and $n \in \mathbb{Z}$. Express $\frac{a}{b}=\frac{1}{315}$ in this form.