(i) State the law of quadratic reciprocity. For $p \neq 5$ an odd prime, evaluate the Legendre symbol

$$\left(\frac{5}{p}\right)$$

(ii) (a) Let $p_{1}, \ldots, p_{m}$ and $q_{1}, \ldots, q_{n}$ be distinct odd primes. Show that there exists an integer $x$ that is a quadratic residue modulo each of $p_{1}, \ldots, p_{m}$ and a quadratic non-residue modulo each of $q_{1}, \ldots, q_{n}$.

(b) Let $p$ be an odd prime. Show that

$$\sum_{a=1}^{p-1}\left(\frac{a}{p}\right)=0$$

(c) Let $p$ be an odd prime. Using (b) or otherwise, evaluate

$$\sum_{a=1}^{p-2}\left(\frac{a}{p}\right)\left(\frac{a+1}{p}\right)$$

$\left[\right.$ Hint for $(c)$ : Use the equality $\left(\frac{x^{2} y}{p}\right)=\left(\frac{y}{p}\right)$, valid when $p$ does not divide $\left.x .\right]$