ELMO SL 2011 N3

Let $n>1$ be a fixed positive integer, and call an $n$ -tuple $(a_1,a_2,\ldots,a_n)$ of integers greater than $1$ good if and only if $a_i\Big|\left(\frac{a_1a_2\cdots a_n}{a_i}-1\right)$ for $i=1,2,\ldots,n$ . Prove that there are finitely many good $n$ -tuples.

Mitchell Lee.