We suppose that polygon is consist of $k$ horizontal vectors and $k$ vertical vectors.Let $k$ horizontal vectors be $\overrightarrow{x_{1}},\dots ,\overrightarrow{x_{k}}$.From $\sum_{i=1}^{k}\overrightarrow{x_{i}}=\overrightarrow{0}$ and $|\overrightarrow{x_{i}}|$ is odd,then $k$ is even.Therefore the number of sides is $2k\equiv 0 \mod4\blacksquare$