The statement of this problem is elementary... How about the proof ? Let $p$ be an odd prime number. For every integer $a$ , define the number
$ S_{a}=\frac{a}{1}+\frac{a^{2}}{2}+....+\frac{a^{p-1}}{p-1} $.
Let $m$ and $n$ be integers such that $ S_{3}+S_{4}-3S_{2}=\frac{m}{n} $. Prove that $p$ divides $m$.