<s>[INST] $2005|(a^3+b^3) , 2005|(a^4+b^4 ) \implies2005|a^5+b^5$ How can I show that if $$2005|a^3+b^3 , 2005|a^4+b^4$$ then $$2005|a^5+b^5$$
I'm trying to solve them from $a^{2k+1} + b^{2k+1}=...$ but I'm not getting anywhere.
Can you please point in me the correct direction?
Thanks in advance
 [/INST] Actually you don't need the $a^4 + b^4$.
$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$. It turns out that $a^2 - ab + b^2 \equiv 0$ has no solutions except $(0,0)$ mod either $5$ or $401$, so the only way to have $a^3 + b^3 \equiv 0 \mod 2005$ is 
$a + b \equiv 0$, and then you also have $a^k + b^k \equiv 0 \mod 2005$ for every odd positive integer $k$.
</s>