I was reading about using modular arithmetic for Diophantine equations, and I stumble upon this problem:
Solve in integers the equation $x^3-3xy+y^3=2005$
The author takes the equation mod 9, so for the left side we have $2005 \equiv -2 (\mod9)$. Then he proceeds to state that for $3\mid xy$ then the left side would be $\equiv 0$ or $\equiv -1 , 3,$ or $4$ $(\mod9)$, and therefore for both cases (I don't know which both cases the author is talking about) the equation is impossible for modulo 9. However I don't understand what is the motivation behind using $3\mid xy$. (I'm aware that if we use that then $3xy$ would be $9k$ and that eliminates a term, but what if we assume $3\nmid$xy ?) And since $a^3 \equiv -1,0,1 (\mod 9)$. Shouldn't the remainders of $x^3+y^3$ be $ -1,0$ or $7$?