Question 178
The function $r_{n}(x)$ is the remainder when $x$ is divided by $n$, where $0 \leq r_{n}(x) \lt n$. For which $n$ does there exist some ordering $\{a_{1}, \ldots , a_{n - 1}\}$ of $\{1, 2, \ldots , n - 1\}$ such that $\{r_{n}(a_{1}), r_{n}(2 \times a_{2}), \ldots , r_{n}((n - 1) \times a_{n - 1})\}$ is an ordering of $\{1, 2, \ldots , n - 1\}$?
(An ordering of $\{1, 2, \ldots , n - 1\}$ is the sequence of numbers 1 to $n - 1$ in some order.)
(An ordering of $\{1, 2, \ldots , n - 1\}$ is the sequence of numbers 1 to $n - 1$ in some order.)