MathDB

Let

m, n \ge 2

be positive integers, and let

a_{1}, a_{2}, \cdots,a_{n}

be integers, none of which is a multiple of

m^{n-1}

. Show that there exist integers

e_{1}, e_{2}, \cdots, e_{n}

, not all zero, with

\vert e_i \vert<m

for all

i

, such that

e_{1}a_{1}+e_{2}a_{2}+ \cdots +e_{n}a_{n}

is a multiple of

m^n

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