We may assume we're working in $\mathbb Z_t$, with the statement of the problem slightly modified to say that for all $x\in\mathbb Z_t$, exactly one of $x+a_i$ is colored. Assume the residues $u_1,\ldots,u_p$ are colored. The number of groups of the form $x+a_1,\ldots,x+a_n$ which contain $u_j$ is the same for all $j$, and each such group (there are $t$ of them, depending only on $x$) contains some $u_j$, which means that we have $t=np$ and we're done.