State the Chinese Remainder Theorem.

Let $N$ be an odd positive integer. Define the Jacobi symbol $\left(\frac{a}{N}\right)$. Which of the following statements are true, and which are false? Give a proof or counterexample as appropriate.

(i) If $\left(\frac{a}{N}\right)=1$ then the congruence $x^{2} \equiv a(\bmod N)$ is soluble.

(ii) If $N$ is not a square then $\sum_{a=1}^{N}\left(\frac{a}{N}\right)=0$.

(iii) If $N$ is composite then there exists an integer a coprime to $N$ with

$$a^{N-1} \not \equiv 1 \quad(\bmod N)$$

(iv) If $N$ is composite then there exists an integer $a$ coprime to $N$ with

$$a^{(N-1) / 2} \not \equiv\left(\frac{a}{N}\right) \quad(\bmod N)$$