Let p, q, r be distinct primes greater than 3, and let n = pqr. Show that if $x \in \mathbb{Z}$ satisfies $x^{2} \equiv  9\mod{n}$ 
then $x \equiv  ±3 \mod{p}$, $x \equiv  ±3 \mod{q}$ and  $x \equiv  ±3 \mod {r}$.
I'm not sure what to do. Any help is appreciated.
Thanks.

Let $x \in \mathbb{Z}$, and $n=pqr$, such that $x^{2} \equiv 9 \mod {n} $, ie,  $\exists k\in \mathbb{Z}$ such that 
\begin{eqnarray}
x^{2}-9 &=& nk\\
(x-3)(x+3)&=&nk
\end{eqnarray}
Now


*

*P.D. $x\equiv 3\mod{p}$
\begin{eqnarray}
(x-3)(x+3)&=&nk\\
(x-3)(x+3)&=&pqrk\\
(x-3)(x+3)&=&p(qrk)\\
x&\equiv&\pm3\mod{p}
\end{eqnarray}

*P.D. $x\equiv 3\mod{q}$
\begin{eqnarray}
(x-3)(x+3)&=&nk\\
(x-3)(x+3)&=&pqrk\\
(x-3)(x+3)&=&q(prk)\\
x&\equiv&\pm3\mod{q}
\end{eqnarray}

*P.D. $x\equiv 3\mod{r}$
\begin{eqnarray}
(x-3)(x+3)&=&nk\\
(x-3)(x+3)&=&pqrk\\
(x-3)(x+3)&=&r(pqk)\\
x&\equiv&\pm3\mod{r}
\end{eqnarray}

