Number Theory Question: $x^2-33y^3=10$ no solutions

Question

I've been struggling to get my head around this for a while!

Show that:

$x^2 - 33y^3 = 10$

has no integral solutions

Answer 1

Check modulo $11$. 10 is not a quadratic residue modulo $11$

Taking the equation modulo $11$ and noting that $33$ is divisble by $11$.

$$x^2\equiv 10\pmod{11}$$