basic number theory - polynomial congruence 
Problem: Determine whether $x^{2} \equiv 5$ mod $120$ has solution. If
  so, how many? 

NOTE: This is a specific question, but is there a method for answering this question given any set of numbers?
Thoughts: Not exactly sure. I want to rearrange the terms to say that this means $x^{2} + 120y = 5$ for some $y \in \mathbb{Z}$. But this doesn't tell me anything either. So...?

The answer is zero solutions.
$x^{2}\equiv{5}\mod{120} \rightarrow x^{2}=120y+5 \rightarrow x^{2}\equiv{5}\mod{8}$
$x^{2}\equiv0,1,4\mod{8}$
Can also be $x^{2}\equiv2\mod{3}$ while quadratic numbers are either $0$ or $1\mod{3}$. Credits: lulu
