BALKAN-SL 2019 N2

Let $S \subset \{ 1, \dots, n \}$ be a nonempty set, where $n$ is a positive integer. We denote by $s$ the greatest common divisor of the elements of the set $S$ . We assume that $s \not= 1$ and let $d$ be its smallest divisor greater than $1$ . Let $T \subset \{ 1, \dots, n \}$ be a set such that $S \subset T$ and $|T| \ge 1 + \left[ \frac{n}{d} \right]$ . Prove that the greatest common divisor of the elements in $T$ is $1$ .

[Second Version] Let $n(n \ge 1)$ be a positive integer and $U = \{ 1, \dots, n \}$ . Let $S$ be a nonempty subset of $U$ and let $d (d \not= 1)$ be the smallest common divisor of all elements of the set $S$ . Find the smallest positive integer $k$ such that for any subset $T$ of $U$ , consisting of $k$ elements, with $S \subset T$ , the greatest common divisor of all elements of $T$ is equal to $1$ .