<s>[INST] Prove $1^{2007}+2^{2007}+\cdots+n^{2007}$ is not divisible by $n+2$ Prove that for any odd natural number $n$, the number $1^{2007}+2^{2007}+\cdots+n^{2007}$ is not divisible by $n+2$.
 [/INST] By taking modulo $n + 2$, and partition the terms as groups of two
each,
$$\begin{align*}
1^{2007}+2^{2007}+3^{2007}+\cdots n^{2007} \\
 \equiv1+\big\lfloor 2^{2007}+(n)^{2007}\big\rfloor+\cdots +\cdots \big\lfloor \ \Big(\frac{n+1}{2}\Big)^{2007}+\Big(\frac{n+3}{2}\Big)^{2007}\big\rfloor\\ \equiv1+\big\lfloor 2^{2007}+(-2)^{2007}\big\rfloor+\cdots +\cdots \big\lfloor \ \Big(\frac{n+1}{2}\Big)^{2007}+\Big(-\frac{n+1}{2}\Big)^{2007}\big\rfloor\\ \equiv 1\pmod{n+2}
\end{align*}
$$
Thus, the conclusion is proven
</s>