$n^{11} $ $mod11$ $n$ fermma theorem

Com10atorics wrote: Prove that $n^{11}-n$ is divisible by $11$. Troll... If $11|n$, then $11|n^{11} - n$ Else , $gcd(n,11)=1$.So by fermat's theorem, $n^{11-1} \equiv 1(mod$ $11)$ $\implies n^{10} \equiv 1(mod$ $11)$ Done!

fermat's theorem.......why do kosovo's 2009 national mo contain so many troll(school?) questions btw ?