Let
![m,n\geq2](/media/m/0/5/1/051dca1e5a565073ab834b966c591f18.png)
be positive integers, and let
![a_1,a_2,\ldots ,a_n](/media/m/3/f/4/3f485c44ad42ff667695792a1098ff6e.png)
be integers, none of which is a multiple of
![m^{n-1}](/media/m/c/f/a/cfaf980e72354cfa3ab5d7d842f5c4f7.png)
. Show that there exist integers
![e_1,e_2,\ldots,e_n](/media/m/8/b/c/8bc67dad167c2fa0e0145548fba7aee3.png)
, not all zero, with
![\left|{\,e}_i\,\right|<m](/media/m/a/3/c/a3c92f3233e53f37c990e3a61293d45d.png)
for all
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
, such that
![e_1a_1+e_2a_2+\,\ldots\,+e_na_n](/media/m/c/d/e/cde6cef84c8df5868f4c8e525d131826.png)
is a multiple of
![m^n](/media/m/6/5/6/656091c558dff45cb142d07ffb312bb4.png)
.
%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 $m^n$.