Let $s \in \mathbb{N}^*$. For $n \in \mathbb{N}$, let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. We denote $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ the gcd of $X_n^{(1)}, \ldots, X_n^{(s)}$. For $r \in \mathbb{N}^*$ and $i \in \{1, 2, \ldots, s\}$, calculate $\mathbf{P}(r \mid X_n^{(i)})$ and show that $\mathbf{P}(r \mid X_n^{(i)}) \leqslant \frac{1}{r}$. Deduce that $$\lim_{n \rightarrow +\infty} \mathbf{P}\left(r \mid Z_n^{(s)}\right) = \frac{1}{r^s}$$
Let $s \in \mathbb{N}^*$. For $n \in \mathbb{N}$, let $X_n^{(1)}, X_n^{(2)}, \ldots, X_n^{(s)}$ be $s$ mutually independent random variables all following the uniform distribution on $\{1, 2, \ldots, n\}$. We denote $Z_n^{(s)} = X_n^{(1)} \wedge \ldots \wedge X_n^{(s)}$ the gcd of $X_n^{(1)}, \ldots, X_n^{(s)}$. For $r \in \mathbb{N}^*$ and $i \in \{1, 2, \ldots, s\}$, calculate $\mathbf{P}(r \mid X_n^{(i)})$ and show that $\mathbf{P}(r \mid X_n^{(i)}) \leqslant \frac{1}{r}$. Deduce that
$$\lim_{n \rightarrow +\infty} \mathbf{P}\left(r \mid Z_n^{(s)}\right) = \frac{1}{r^s}$$