Question 85
Let $p \ge 5$ be a prime number and $a$ be an integer with $1 \lt a \lt p-1$. Let $a_1, a_2, \dots, a_n$ be all possible remainders that powers of $a$ (that is, $1, a, a^2, a^3, \dots$) can leave when divided by $p$. If the remainders $a_1, a_2, \dots, a_n$ form an arithmetic sequence modulo $p$ in some order, prove that $n = p-1$. (Note that, for example, 6, 1 and 3 form an arithmetic progression modulo 7.)