MathNet 1994 — Part 01 Question 30

Let $p \gt 2$ be a prime number and $1 + \frac{1}{2^{3}} + \frac{1}{3^{3}} + \cdots + \frac{1}{(p-1)^{3}} = \frac{m}{n}$ where $m$ and $n$ are relatively prime. Show that $m$ is a multiple of $p$.