Let be positive integers, and let be integers, none of which is a multiple of . Show that there exist integers , not all zero, with for all , such that is a multiple of .
%V0
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 $m^n$.
Izvor: Međunarodna matematička olimpijada, shortlist 2002
Školjka 
be positive integers, and let 
be integers, none of which is a multiple of 
. Show that there exist integers 
, not all zero, with 
for all 
, such that 
is a multiple of 
.