7. Let $a$ and $b=r_{0}$ be relatively prime odd positive integers such that
$$\begin{array}{l} a=r_{0} q_{1}+\epsilon_{1} r_{1} \\ r_{0}=r_{1} q_{2}+\epsilon_{2} r_{2} \\ \cdot \\ \cdot \\ r_{n-1}=r_{n-1} q_{n-1}+\epsilon_{n} r_{n} \end{array}$$
where $q_{i}$ is a nonnegative even integer, $\epsilon_{i}= \pm 1, r_{i}$ is a positive integer with $r_{i}<r_{i-1}$, for $i=1,2, \ldots, n_{j}$, and $r_{n}=1$. These equations are obtained by successively using the modified division algorithm given in problem 10 of Section 1.2 .
a) Show that the Jacobi symbol $\left(\frac{a}{b}\right)$ is given by
$$\left(\frac{a}{b}\right)=(-1)^{\left(\frac{r_{0}-1}{2} \frac{r_{1}-1}{2}+\frac{r_{1}-1}{2} \frac{\epsilon_{r}-1}{2}+\cdots+\frac{r_{t-1}-1}{2} \cdot \frac{r_{t}^{r}-1}{2}\right)}$$
b) Show that the Jacobi symbol $\left(\frac{a}{b}\right)$ is given by
$$\left(\frac{a}{b}\right)=(-1)^{T}$$
where $T$ is the number of integers $i, 1 \leqslant i \leqslant n$, with $r_{i-1} \equiv \epsilon_{i} r_{i} \equiv 3$ $(\bmod 4)$.