ISL 2002 N5

Let $m,n\geq2$ 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 $\left|{ e}_i\,\right|<m$ for all $i$ , such that $e_1a_1+e_2a_2+\,\ldots\,+e_na_n$ is a multiple of $m^n$ .

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