One of my students asked me the following question. It seemed easy to answer but in fact, I am stucK.

The question: does there exist an infinite subset $S$ of $\mathbb N$ such there exists a positive integer $n_0$ such for all prime $p\ge n_0$ one has
$$\mathrm{Card}(S \pmod p)<p$$
where $S\pmod p$ is the subset of classes of $\mathbb Z/p\mathbb Z$ having a representative in $S$.

Thanks in advance (for me and my student :D )