Now I'm not sure why the following holds:
If $x^2 \equiv a \pmod n$ for some $x \in \mathbb Z$, then $x^2 \equiv a \pmod {p_i}$ for all $i$, where $n=p_1^{t_1} \dots p_r^{t_r}$.
I know that if $a \equiv b \pmod {p_1 \dots p_r}$, then $a \equiv b \pmod {p_i}$ for all $i$, if $p_1, \dots, p_r$ are pairwise relative primes.