Proof Writing & Problem Solving · Superchallenge problem

For how many $n$ with $1 \le n \le 100$ does there EXIST an ordering $a_1, a_2, \ldots, a_n$ of the numbers $1, 2, \ldots, n$ whose partial sums $a_1,\ a_1 + a_2,\ \ldots,\ a_1 + a_2 + \cdots + a_n$ leave $n$ different remainders when divided by $n$?