Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Only for $m=2$, $\sum_{i=1}^mx^{2n+1}_k$ is always divisible by $\sum_{i=1}^mx_k$. Indeed,
$$x+y | x^{2n+1}+y^{2n+1}, n \in \mathbb N.$$
