Prove that for any positive integer $n$, there exist $n$ consecutive positive integers $a_1, a_2,...,a_n$ such that $p_i |a_i $ for each $i$, where $p_i$ denotes the $i$th prime?
Having a lot of trouble with this problem, I tried searching this question up but only got problems similar to it and it still didn't quite help. We are currently covering a section on The Chinese Remainder Theorem. Our teacher showed us two ways of utilizing this theorem, one via back-substitution and one via the classical approach. Looking through my notes I don't see anything resembling this problem, any help is really appreciated.