system of congruence - my approach We have:
$$k^3 + l^3 \equiv 0 \pmod{17}\\
k^2 + l^2 \equiv 0 \pmod{17} $$
And I get:
$$k = 17n+r_k\\
l = 17m+r_l$$
And I analyzed possible rests respect to system of congruences.
My result is: 
$$ k,l \equiv 0\pmod{17}$$
But is it properly ? Is my way is ok ? Is exists other, better solution ?

When we are working with a prime modulus, we can use the fact that $p|ab$ implies that either $p|a$ or $p|b$. In terms of modular arithmetic this is the same as $ab\equiv 0 \mod p$ implies $a\equiv 0$ or $b\equiv 0$. Here we can use this observation to avoid having to deal with lots of cases.
Note that the first equation is equivalent to: $$(k+l)(k^2-kl+l^2)\equiv 0\mod 17$$
Because $17$ is prime we have either $k+l\equiv 0$ or $k^2-kl+l^2\equiv 0$
In the first case we use the second (original) congruence as well and find $(k+l)^2-k^2-l^2=2kl \equiv 0$ so that $kl\equiv 0$
In the second case $k^2-kl+l^2-k^2-l^2=-kl \equiv 0$
In either case $kl\equiv 0$ so one of $k,l$ must be divisible by $17$ - and it is easy to see that this implies both must be multiples of $17$.
