Question 6
Let $a_{1} \lt a_{2} \lt a_{3} \lt \cdots$ be positive integers such that $a_{k+1}$ divides $2(a_{1} + a_{2} + \cdots + a_{k})$ for every $k \geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_{k}$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_{k}$.