SCHWEITZER 2003 Problem 2

Let $p$ be a prime and let $M$ be an $n\times m$ matrix with integer entries such that $Mv\not\equiv 0\pmod{p}$ for any column vector $v\neq 0$ whose entries are $0$ are $1$ . Show that there exists a row vector $x$ with integer entries such that no entry of $xM$ is $0\pmod{p}$ .

(translated by L. Erdős)