grandes-ecoles 2022 Q21

grandes-ecoles · France · x-ens-maths2__mp Number Theory GCD, LCM, and Coprimality

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $s, n \in \mathbb{N}^*$ with $2 \leqslant s \leqslant n$. We randomly draw $s$ numbers from $\{1, 2, \ldots, n\}$ and we denote $P_n(s)$ the probability that these numbers are coprime. Show that $$\lim_{n \rightarrow +\infty} P_n(s) = \frac{1}{\zeta(s)}$$ and give the value of $\lim_{n \rightarrow +\infty} P_n(s)$ in the case where $s = 2$, then $s = 4$, and finally $s = 6$.

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $s, n \in \mathbb{N}^*$ with $2 \leqslant s \leqslant n$. We randomly draw $s$ numbers from $\{1, 2, \ldots, n\}$ and we denote $P_n(s)$ the probability that these numbers are coprime. Show that
$$\lim_{n \rightarrow +\infty} P_n(s) = \frac{1}{\zeta(s)}$$
and give the value of $\lim_{n \rightarrow +\infty} P_n(s)$ in the case where $s = 2$, then $s = 4$, and finally $s = 6$.

Paper Questions

Q1a Q1b Q1c Q1d Q2a Q2b Q3a Q3b Q4 Q5a Q5b Q6 Q7a Q7b Q7c Q8a Q8b Q9 Q10a Q10b Q10c Q11 Q12a Q12b Q12c Q12d Q12e Q13 Q14 Q15a Q15b Q15c Q16 Q17a Q17b Q17c Q18 Q19 Q20a Q20b Q21