<s>[INST] Solve system of congruences $k^2 + l^2 \equiv 0 \mod 17 \wedge k^3+l^3 \equiv 0 \mod 17$ 
Solve system of congruences 
  $$k^2 + l^2 \equiv  0 \mod 17 \wedge k^3+l^3 \equiv 0 \mod 17$$

Is there any faster way to solve that congurence than looking at table and finding such pairs of $i$ that in both cases it gives $0$? When we have this table it is very fast job but making it is... time-consuming
\begin{array}{|c|c|c|c|}
\hline
i^2 \mod 17 & i^3 \mod 17 \\ \hline
0 & 0 \\ \hline
1 & 1 \\ \hline
4 & 8 \\ \hline
9 & 10 \\ \hline
16 & 13 \\ \hline
8 & 6 \\ \hline
2 & 12 \\ \hline
15 & 3 \\ \hline
13 & 2 \\ \hline
13 & 15 \\ \hline
15 & 14 \\ \hline
2 & 5 \\ \hline
8 & 11 \\ \hline
16 & 4 \\ \hline
9 & 7 \\ \hline
4 & 9 \\ \hline
1 & 16 \\ \hline
0 & 0 \\ \hline
1 & 1 \\ \hline
4 & 8 \\ \hline
\end{array}
 [/INST] From $k^3 + l^3 \equiv 0 \pmod{17}$, we have
$$k^3 \equiv (-l)^3 \pmod {17}.$$
But, as you can see from your table, cubes are distinct modulo $17$, which effectively lets us take cube roots: thus, $k \equiv -l \pmod {17}$. Therefore
$$k^2 + (-k)^2 \equiv 0 \pmod {17},$$
so $2k^2 \equiv 0 \pmod {17}$. Divide by $2$ and conclude that $k \equiv l \equiv 0 \pmod{17}$.
</s>