Question 1
Let $p$ be an odd prime such that $2^h\not\equiv1\pmod p$ for every integer $1\le h\lt p-1$. Let $a$ be an even integer with $p/2\lt a\lt p$. Define $a_0=a$ and $a_{n+1}=p-b_n$ for $n\ge0$, where $b_n$ is the greatest odd divisor of $a_n$. Prove that $(a_n)$ is periodic and determine its least positive period.