If $k$ is an odd positive integer, prove that for any integer $ n\geq 1$, $(1^k+2^k+3^k+ \ldots +n^k)$ is divisible by $ (1+2+3+ \ldots +n)$.
I didn't get how it is connected( or different) than this: suppose that $n$ is natural number and even, show that $ n∤1^n+2^n+3^n+…(n−1)^n$
I tried for small numbers $n=3,4$ then use induction on $k$ but how to show for any natural $n$? does the answer in the link provides an insight to the solution.
Hints would suffice. Thanks