Question 25
Let $p$ be an odd prime number. For every integer $a$, define the number
$$
S_{a} = \frac{a}{1} + \frac{a^{2}}{2} + \cdots + \frac{a^{p-1}}{p-1}.
$$
Let $m$ and $n$ be integers such that
$$
S_{3} + S_{4} - 3 S_{2} = \frac{m}{n}
$$
Prove that $p$ divides $m$.
$$
S_{a} = \frac{a}{1} + \frac{a^{2}}{2} + \cdots + \frac{a^{p-1}}{p-1}.
$$
Let $m$ and $n$ be integers such that
$$
S_{3} + S_{4} - 3 S_{2} = \frac{m}{n}
$$
Prove that $p$ divides $m$.