quadratic reciprocity

Question

I know $x^2\equiv-7\pmod7$ has solutions. How can I check if $x^2\equiv-7\pmod{49}$ has solutions? I know $-7\equiv42\pmod{49}$ but $49$ isn't a prime so I can't use Euler's criterion. How shall I do this other than check the squares of all numbers modulo $49$?

Answer 1

Note that if $y \equiv -7 \mod 49$, $y$ is divisible by 7, but not by 49, so $y$ is not a square.

Answer 2

If $x^2\equiv-7\pmod{49}$, then $7|x$, hence $49|x^2$, which means $$ x^2\equiv 0\pmod{49} $$ a contradiction