NL-NWO 1972 Problem 4

On a circle with radius $1$ the points $A_1, A_2,..., A_n$ lie such that every arc $A_iA_{i+i}$ has length $\frac{2\pi}{n}= a$ . Given that there exists a set $V$ consisting of $k$ of these points ( $k < n$ ), which has the property that each of the arc lengths $a$ , $2a$ $,...$ , $(n- 1)a$ can be obtained in exactly one way be taken as the length of an arc traversed in a positive sense, beginning and ending in a point of $V$ . Express $n$ in terms of $k$ and give the set $V$ for the case $n = 7$ .