Question 136
Let $m \gt 1$ be an integer. It is known that there exists a prime number in the interval $[2m - \sqrt{m} + 1, 2m]$. Prove that among any $m$ pairwise distinct positive integers $a_1, a_2, \dots, a_m$, there exist two numbers $a_i$ and $a_j$ ($1 \le i, j \le m$) such that
$$
\frac{a_i}{(a_i, a_j)} \ge m,
$$
where $(a_i, a_j)$ denotes the greatest common divisor of the positive integers $a_i$ and $a_j$.
$$
\frac{a_i}{(a_i, a_j)} \ge m,
$$
where $(a_i, a_j)$ denotes the greatest common divisor of the positive integers $a_i$ and $a_j$.