Prime power residue and linear coverings of vector space over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math>

Let $q$ be an odd prime and $B = \{b_{j}\}_{j=1}^{l}$ a finite set of nonzero integers that does not contain perfect $q^{th}$ power. We show $B$ has power modulo every $p \neq q$ dividing $\prod_{b\in B} b$ if only corrresponds to linear hyperplane covering $\mathbb{F}_{q}^{k}$. Here, $k$ is the number distinct factors $q$-free part elements $B$. Consequently: $(i)$ \subset\mathbb{Z}\setminus\{0\}$ with cardinality less than $q+1$ cannot have almost unless it contains $(ii)$ For \{b_{j}\}_{j=1}^{l} for $\big(c_{j}\big)_{j=1}^{l} \in\Big(\mathbb{F}_{q}\setminus\{0\}\Big)^{l}$ $\prod_{j=1}^{l}$ $\{b_{j}^{c_{j}}\}_{j=1}^{l}$ so.