Question 139
Let $p$ be an odd prime. Find all possible remainders modulo $p^2$ of sums of the form
$$
a^{p-1} + a^{p-2}b + \dots + ab^{p-2} + b^{p-1}
$$
for integers $a$ and $b$.
$$
a^{p-1} + a^{p-2}b + \dots + ab^{p-2} + b^{p-1}
$$
for integers $a$ and $b$.