Question 4
Find the smallest positive integer $n$, or show no such $n$ exists, such that one can find infinitely many distinct $n$-tuplets of positive rationals $(a_1,a_2,\ldots,a_n)$ satisfying
$$\sum_{i=1}^n a_i\in\mathbb Z,\qquad\sum_{i=1}^n\frac1{a_i}\in\mathbb Z.$$
$$\sum_{i=1}^n a_i\in\mathbb Z,\qquad\sum_{i=1}^n\frac1{a_i}\in\mathbb Z.$$