Quite a trivial question in comparison to the ones found on here, but here it goes:
"By applying (mod 5) to both equations, show that the system of equations below has no integer solutions:
$11x^2 + 5xy = 7 $
$9x^2 + 10y^3 = −3$ "
Now, the solution jumps straight to:
$x^2 = 2 \pmod 5$
$-x^2 = -3 \pmod 5. $
I'm struggling to figure out how exactly they get to that stage, and at that, how these last two equations imply there are no integer solutions.
$11x^2+5xy=7\implies x^2\equiv2\pmod5$, because $11\equiv1\pmod5, $
$5\equiv0\pmod5$, and $7\equiv2\pmod5$.
$9x^2+10y^3=-3\implies -x^2\equiv-3\pmod5\iff x^2\equiv 3\pmod5$,
because $9\equiv-1\pmod5$ and $ 10\equiv0\pmod5$.
There are no solutions, because $2\equiv x^2\equiv3\pmod5$ is a contradiction.