Question 17
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 $(a_1, a_2, \dots, a_n)$ satisfying:
$$
\sum_{i=1}^{n} a_i \in \mathbb{Z}, \quad \sum_{i=1}^{n} \frac{1}{a_i} \in \mathbb{Z}.
$$
$$
\sum_{i=1}^{n} a_i \in \mathbb{Z}, \quad \sum_{i=1}^{n} \frac{1}{a_i} \in \mathbb{Z}.
$$