First question: Let $a$, $b$, $c$ be distinct primes larger than $3$, and let $x = abc$. Show that if $p$ is a positive integer and $p^2 \equiv 9 \pmod {x}$, then $p \equiv \pm 3 \pmod {a}$, $p \equiv \pm 3 \pmod{b}$, and $p \equiv \pm 3 \pmod{c}$.
Second question: Using the fact that $1001 = (7)(11)(13)$ find the eight solutions to $y^2 \equiv 9 \pmod{1001}$ in the set $\{ 1, 2, 3,\ldots, 1000 \}$.