Question 29
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 $m$.