Question 26
Find the smallest positive integer $n$ satisfying the following condition, or prove that no positive integer $n$ satisfies it:
there exist infinitely many $n$-tuples $(a_1, a_2, \cdots, a_n)$ of $n$ distinct positive rational numbers such that
$$
a_1 + a_2 + \cdots + a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}
$$
are both integers.
there exist infinitely many $n$-tuples $(a_1, a_2, \cdots, a_n)$ of $n$ distinct positive rational numbers such that
$$
a_1 + a_2 + \cdots + a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}
$$
are both integers.