Problem 467

Erdős Problem #467: declared status 'open'. Formalized: no. Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\leq x$ and a decomposition $\{p\leq x\}=A\sqcup B$ into two non-empty sets such that, for all $n<x$, there exist some $p\in A$ and $q\in B$ such that $n\equiv a_p\pmod{p}$ and $n\equiv a_q\pmod{q}$. Prize: no. Tags: number theory.