Proof for Elementary Number Theory

Prove that for any positive integer $n$, there exist $n$ consecutive positive integers $a_1, a_2,...,a_n$ such that $p_i$ divides $a_i$ for each $i$, where $p_i$ denotes the $i$-th prime.

I'm not sure how to prove this. Could we possibly use the Chinese Remainder theorem? If so how?