Prove that $3x^3- 7y^3+21 z^3=2$ has no integer solutions.

Question

I have this math problem I'm kind of stuck on.

Prove that $3x^3- 7y^3+21 z^3=2$ has no integer solutions.

The reason I am stuck on this problem is because I am supposed to be using modular arithmetic to prove it. I'm not really sure where to start. Thanks.

Answer 1

Use mod $7$ to find that $3x^3\equiv 2\pmod{7}$. But by Fermat's Little theorem $x^6\equiv \{0,1\}\pmod{7}$, so $x^3\equiv \{0,\pm 1\}\pmod{7}$, so $3x^3\equiv \{0,\pm 3\}\equiv \{0,3,4\}\not\equiv 2\pmod{7}$.