Question 5
Let $m, n \geq 2$ be positive integers, and let $a_1, a_2, \ldots, a_n$ be integers, none of which is a multiple of $m^{n-1}$. Show that there exist integers $e_1, e_2, \ldots, e_n$, not all zero, with $|e_i| \lt m$ for all $i$, such that $e_1 a_1 + e_2 a_2 + \ldots + e_n a_n$ is a multiple of $m^n$.