  For coprime positive integers $q$ and $e$, let $m(q,e)$ denote the least
positive integer $t$ such that there exists a sum of $t$ powers of $q$ which is
divisible by $e$. We prove an upper bound for $m(q.e)$ and investigate the case
where $m(q,e)$ is "large". We also pay special attention to the situation where
$e$ is a prime power.
