$2005|(a^3+b^3) , 2005|(a^4+b^4 ) \implies2005|a^5+b^5$ 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

If prime $p|(a^3+b^3), p|(a+b)(a^3+b^3) \implies p| \{(a^4+b^4)+ab(a^2+b^2)\}$
If $p|(a^4+b^4), p$  must divide $ab(a^2+b^2)$
If $p|a,$ $p$ must divide $b$ as $p|(a^3+b^3)$ 
If $p|a$ and $p|b,$ then $p|(a^n+b^n)$ for integer $n\ge1$
Else $p\not\mid ab $  and  $p|(a^2+b^2)\implies p|(a+b)(a^2+b^2) \implies p| \{(a^3+b^3)+ab(a+b)\}$
As $p|(a^3+b^3),p|ab(a+b)\implies p|(a+b)$ as $p\not\mid ab$
As $p|(a+b)$ then $p|(a^m+b^m)$ for odd integer $m\ge1$
Now put $p=5,p=401$ separately and use lcm$[5,401]=2005$ 
