<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] $(a^5+b^5) = (a+b)(a^4+b^4) - ab(a^3+b^3)$.
So for any $n$, if $n \mid a^3+b^3$ and $n \mid a^4+b^4$ then $n \mid a^5+b^5$.
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