<s>[INST] Number of solutions to the congruence $x^2\equiv 121\pmod {1800}$ I'm trying to find the number of solutions to this congruence:
 $$x^2\equiv 121\pmod {1800}$$
I thought about writing it as a system of congruences. As $1800=3^2 \cdot 5^2 \cdot 2^3$, we get:
$x^2\equiv 121\pmod {5^2} \;,\; x^2\equiv 121\pmod {3^2} \;,\; x^2\equiv 121\pmod {2^3}$
$\Downarrow$
$ x^2\equiv 21\pmod {5^2} \;,\; x^2\equiv 4\pmod {3^2} \;,\; x^2\equiv 1\pmod {2^3} $
Now I'm pretty stuck with how to solve each of the above quadratic congruences, is there a fast way to do it? 
Thanks
 [/INST] No, there's not a fast way.  To solve $x^2\equiv a \pmod{p^n}$ we first solve $x^2\equiv a \pmod{p}$ and then use Hensel's Lemma to "lift" the solution to $\pmod{p^2}$ and then $\pmod{p^3}$, etc.  Sometimes the solutions lift uniquely, sometimes not.  
In the case of your problem, there are two solutions modulo $5$ which lift uniquely to two solutions modulo $25$.  The exact same thing happens modulo $9$.
There is one solution modulo $2.$  It lifts to two solutions modulo $4.$  Each of these in turn lifts to two solutions modulo $8$, for a total of 4 solutions modulo $8.$
So in the end, there are $2\times 2\times 4 = 16$ solutions modulo $1800.$
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