Let $p$ be a prime number and $k<p$ a positive integer. Let $m=\left\lceil \frac{p}{k+1}\right\rceil$. Show that there is a set $A\subseteq\{1,2,\ldots, p-1\}$ with at most 2m elements such that for every $a\in\{1,2,\ldots,p-1\}$ there are $b\in A$ and $c\in\{1,2,\ldots,k\}$ such that $p$ divides $a-bc$.