Let $n_1<n_2<\cdots$ be an infinite sequence of integers with associated $a_k(\mathrm{mod}{n}_k)$, such that for some $ϵ>0$ we have $n_k>(1+ϵ)k\log k$ for all $k$. Then$$\#\{m<n_k:m\not\equiv a_i(\mathrm{mod}{n}_i)\text{ for }1\le i\le k\}\ne o(k).$$