Disprove the following:
(i) If $\gcd (x, n) = 1$ and $x^2 ≡ 1 \pmod n$ then $x ≡ ±1 \pmod n$.
(ii) If $\gcd (x, n) = 1$ and $x^2 ≡ a \pmod n$ then $x ≡ ±a \pmod n$.
We'll do this by counter-example.
(i) Using $x = 2, n = 3$, we have $\gcd(2,3)=1$ and $2^2 \equiv 1 \pmod 3$. But $2 \equiv \pm 1 \pmod 3$ is false.
(ii) Using $x = 3, n = 5$, we have $\gcd(3,5)=1$ and $3^2 \equiv 4 \pmod 5$. But $3 \equiv \pm 4 \pmod 5$ is false.