Question 142
**Prove the following statements:**
(1) If $2n-1$ is a prime number, then for any group of distinct positive integers $a_1, a_2, \dots, a_n$ there exist $i, j \in \{1, 2, \dots, n\}$ such that $\frac{a_i+a_j}{(a_i, a_j)} \ge 2n-1$.
(2) If $2n-1$ is a composite number, then there exists a group of distinct positive integers $a_1, a_2, \dots, a_n$ such that $\frac{a_i+a_j}{(a_i, a_j)} \lt 2n-1$ for any $i, j \in \{1, 2, \dots, n\}$.
Here $(x, y)$ denotes the greatest common divisor of positive integers $x$ and $y$.
(1) If $2n-1$ is a prime number, then for any group of distinct positive integers $a_1, a_2, \dots, a_n$ there exist $i, j \in \{1, 2, \dots, n\}$ such that $\frac{a_i+a_j}{(a_i, a_j)} \ge 2n-1$.
(2) If $2n-1$ is a composite number, then there exists a group of distinct positive integers $a_1, a_2, \dots, a_n$ such that $\frac{a_i+a_j}{(a_i, a_j)} \lt 2n-1$ for any $i, j \in \{1, 2, \dots, n\}$.
Here $(x, y)$ denotes the greatest common divisor of positive integers $x$ and $y$.