VN TST 1999 Question 6

Let $p>2$ be prime. A monkey sits at every vertex of a $p$-gon. The source says “$p$ beans are given to each monkey”, under the following rule: give the first bean to an arbitrary monkey, the next to the monkey at the third vertex clockwise from that monkey, the third to the monkey at the fifth vertex clockwise from the previous recipient, and so on. In general the $(k+1)$st bean goes to the monkey at the $(2k+1)$st vertex clockwise from the recipient of the $k$th bean.
1. How many monkeys receive no beans?
2. How many edges have monkeys receiving beans at both endpoints?