Is there such an integer? $\{a_n\}_{n=1}^{\infty}$ and $\{b_n\}_{n=1}^{\infty}$ are two infinite sequences of natural numbers such that for all $n$, $0\leq b_n<a_n$. Is it possible to exist an integer $k$ such that for all $n$, $k\equiv b_n(\mbox{mod } a_n)$?
Thanks for your answer!

Yes, it is possible if we take $b_n=k$ for all $n$ and $a_n$ to be anything greater than $k$.
