RIOPLATENSE 2018 Problem 5

Let $n$ be a positive integer. Find all $n$ - rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$\frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$ is integer for every integer $d\ge 0$