From 1b954d07b72a1999476cb80458b84bf054587795 Mon Sep 17 00:00:00 2001
From: couchot <jf.couchot@gmail.com>
Date: Tue, 17 Dec 2013 09:35:11 +0100
Subject: [PATCH] ajout de partiels
---
partiels/131217S1/partiel.aux | 16 +
partiels/131217S1/partiel.log | 1144 +++++++++++++++++++++++++++++
partiels/131217S1/partiel.pdf | Bin 0 -> 105184 bytes
partiels/131217S1/partiel.tex | 187 +++++
partiels/131217S1/partiel.tex.bak | 193 +++++
partiels/131217S1/partiel.tex~ | 193 +++++
partiels/131217S1/partiel.thm | 0
partiels/131217S1/symboles.sty | 1 +
8 files changed, 1734 insertions(+)
create mode 100644 partiels/131217S1/partiel.aux
create mode 100644 partiels/131217S1/partiel.log
create mode 100644 partiels/131217S1/partiel.pdf
create mode 100644 partiels/131217S1/partiel.tex
create mode 100644 partiels/131217S1/partiel.tex.bak
create mode 100644 partiels/131217S1/partiel.tex~
create mode 100644 partiels/131217S1/partiel.thm
create mode 120000 partiels/131217S1/symboles.sty
diff --git a/partiels/131217S1/partiel.aux b/partiels/131217S1/partiel.aux
new file mode 100644
index 0000000..b5155d8
--- /dev/null
+++ b/partiels/131217S1/partiel.aux
@@ -0,0 +1,16 @@
+\relax
+\catcode `:\active
+\catcode `;\active
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+\catcode `?\active
+\select@language{french}
+\@writefile{toc}{\select@language{french}}
+\@writefile{lof}{\select@language{french}}
+\@writefile{lot}{\select@language{french}}
+\@writefile{toc}{\contentsline {section}{\numberline {1}(5 pts) Alg\IeC {\`e}bre de Boole }{1}}
+\@writefile{toc}{\contentsline {section}{\numberline {2}(4 pts) Relations binaires entre ensembles}{1}}
+\@writefile{toc}{\contentsline {section}{\numberline {3}(3 pts) D\IeC {\'e}monstration par r\IeC {\'e}currence}{1}}
+\@writefile{toc}{\contentsline {section}{\numberline {4}(7 pts) Plus Grand Commun Diviseur}{2}}
+\newlabel{itm:Q1}{{2}{2}}[None]
+\newlabel{itm:Q2}{{3}{2}}[None]
+\@writefile{toc}{\contentsline {section}{\numberline {5}(5 pts) Nombres (premiers ou pas) de Fermat}{2}}
diff --git a/partiels/131217S1/partiel.log b/partiels/131217S1/partiel.log
new file mode 100644
index 0000000..81944b3
--- /dev/null
+++ b/partiels/131217S1/partiel.log
@@ -0,0 +1,1144 @@
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diff --git a/partiels/131217S1/partiel.tex b/partiels/131217S1/partiel.tex
new file mode 100644
index 0000000..8dd4410
--- /dev/null
+++ b/partiels/131217S1/partiel.tex
@@ -0,0 +1,187 @@
+\documentclass[11pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage[amsmath,thmmarks,thref,framed]{ntheorem}
+\usepackage[dvips]{graphics}
+\usepackage{epsfig}
+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{times}
+\usepackage{tabularx}
+\usepackage{textcomp}
+\usepackage{pst-all}
+\usepackage[a4paper]{geometry}
+
+\input{symboles.sty}
+\geometry{hmargin=1.5cm, tmargin=1.5cm, bmargin=1.5cm}
+
+\title{
+DUT d'informatique. Partiel de mathématiques discrètes.\\
+Semestre 1 (Décembre 2013). Durée 1h30.}
+
+\date{}
+
+\begin{document}
+\maketitle
+
+%\vspace{-3em}
+\noindent Seule une fiche manuscrite de format A5 est autorisée.
+Tous les moyens de communications sont interdits.
+Tous les exercices de ce sujet sont indépendants les uns des autres.
+Dans tout ce qui suit, on considère une algèbre de Boole
+munie des opérateurs classiques ``+'', ``.'',
+``$\overline{\begin{array}{l}~\end{array}}$'' et des variables booléennes
+$a$, $b$, $c$, $d$.
+
+
+%\vspace{-1em}
+\section{(5 pts) Algèbre de Boole }
+On cherche à analyser les codes binaires sur 4 bits.
+
+
+\begin{enumerate}
+\item Dans cette question, un code est correct s'il contient au
+plus deux 1 consécutifs. Par exemple $1010$ est dit correct
+(il ne contient pas de 1 consécutifs) tandis
+$0111$ ne l'est pas (il contient trois 1 consécutifs).
+Le but de cette question est de concevoir la fonction $F$ qui renvoie 1
+si le code de 4 bits $abcd$ est correct et 0 sinon.
+\begin{enumerate}
+\item Ãcrire la table de vérité de cet analyseur dans une table de Karnaugh.
+\item Donner l'expression la plus simplifiée de la fonction $F$.
+\end{enumerate}
+
+\item Dans cette question, un code est correct
+s'il ne contient pas de 0 consécutifs (par exemple $1011$).
+Le but de cette question est de concevoir la fonction $G$ qui renvoie 1
+si le code de 4 bits $abcd$ est correct et 0 sinon.
+\begin{enumerate}
+\item Ãcrire la table de vérité de cet analyseur dans une table de Karnaugh.
+\item Donner l'expression la plus simplifiée de la fonction $G$.
+\end{enumerate}
+
+\item Dans cette question, un code est correct
+s'il vérifie les deux contraintes précédentes.
+Le but de cette question est de concevoir la fonction $H$ qui renvoie 1
+si le code de 4 bits $abcd$ est correct et 0 sinon.
+\begin{enumerate}
+\item Sans utiliser de table de Karnaugh, résoudre algébrique de ce
+problème.
+\item Exprimer $H$ sous la forme d'une somme de trois monômes.
+\item Comment pourrait-on vérifier le résultat sur les tables de Karnaugh
+remplies aux deux premières questions.
+\end{enumerate}
+\end{enumerate}
+
+%\vspace{-1em}
+\section{(4 pts) Relations binaires entre ensembles}
+
+
+\begin{enumerate}
+\item
+Soit $P^*$ l'ensemble des nombres premiers impairs. On considère
+la relation $\mathcal{R}$ entre deux éléments de $P^*$ définie par:
+$$
+p \mathcal{R} q \textrm{ si et seulement si }
+\dfrac{p+q}{2} \textrm{ appartient à $P^*$.}
+$$
+Montrer qu'elle n'est pas transitive.
+
+\item On considère
+la relation $\mathcal{S}$ entre deux éléments de $\Z$ définie par:
+$$
+m \mathcal{S} n \textrm{ si et seulement si }
+m^2 + m = n^2 + n.
+$$
+\begin{enumerate}
+\item Montrer que c'est une relation d'équivalence.
+\item Construire la classe de 0, nommée $\dot{0}$.
+\item Pour tout entier relatif $m$, construire la classe de $m$,
+ nommée $\dot{m}$.
+\end{enumerate}
+\end{enumerate}
+
+%\vspace{-1em}
+\section{(3 pts) Démonstration par récurrence}
+On considère la suite $(U_n)_{n \in \N}$ définie pour
+tout $n \in \N$ par $U_n = 10^n.(9n-1)+1$.
+\begin{enumerate}
+\item Calculer $U_0$, $U_1$ et $U_2$.
+\item Montrer que pour tout $n \in \N$ par
+$U_{n+1} = U_n + 9^2.10^n.(n+1)$.
+\item Montrer que $U_n$ est divisible par 81 pour tout $n \in \N$.
+\end{enumerate}
+
+%\vspace{-1em}
+\section{(7 pts) Plus Grand Commun Diviseur}
+Dans ce qui suit, $n$ est un entier naturel strictement positif.
+On considère trois suites
+$(A_n)_{n \in \N^*}$, $(B_n)_{n \in \N^*}$ et $(d_n)_{n \in \N^*}$
+définies par
+$A_n = n^2 - 2n +2$,
+$B_n = n² +2n +2$
+ et
+$d_n= A_n \land B_n$.
+
+\begin{enumerate}
+\item Calculer $A_1$, $A_2$, $A_3$, $A_4$, $B_1$, $B_2$, $B_3$, $B_4$, $d_1$, $d_2$, $d_3$ et $d_4$. Quelles intuitions avez vous à propos de $d_n$?
+\item \label{itm:Q1} Montrer que si un entier naturel
+ $d$ divise à la fois $A_n$ et $n$, alors $d$ divise 2.
+\item \label{itm:Q2} Montrer que si un entier naturel
+ $d$ divise à la fois $A_n$ et $B_n$, alors $d$ divise $4n$.
+\item Dans cette question on suppose que $n$ est impair.
+\begin{enumerate}
+\item Montrer que $A_n$ et $B_n$ sont impairs.
+ En déduire que $d_n$ est impair.
+\item En utilisant entre autre la question~\ref{itm:Q2}, montrer que $d_n$ divise $n$.
+\item En utilisant entre autre la question~\ref{itm:Q1}, en déduire que $d_n$ divise 2,
+ puis que $A_n$ et $B_n$ sont premiers entre eux.
+\end{enumerate}
+\item On suppose maintenant que $n$ est pair.
+\begin{enumerate}
+\item Montrer que $d_n$ est pair.
+\item Montrer que 4 ne divise pas $A_n$. En déduire la même chose pour $B_n$
+\item Montrer que $d_n$ peut s'écrire sous la forme $d_n = 2.p$,
+ où $p$ est impair.
+\item En utilisant entre autre la question~\ref{itm:Q2}, en déduire
+ que $p$ divise $n$.
+\item En utilisant entre autre la question~\ref{itm:Q1}, en déduire que $d_n=2$.
+\end{enumerate}
+\end{enumerate}
+
+
+
+%\vspace{-1em}
+\section{(5 pts) Nombres (premiers ou pas) de Fermat}
+Pour $p \in \N$, les nombres de Fermat sont ceux de la forme
+$F_p = 2^{2^p}+1$.
+
+
+\begin{enumerate}
+\item
+Montrer que si $n$ n'est pas une puissance de 2, alors
+$2^n+1$ n'est pas un nombre premier.
+On pourra se servir de l'égalité
+$$
+x^n+ 1 = (x+1)(x^{n-1}-x^{n-2}+\ldots - x + 1)
+$$ qui est est consiédérée comme admise
+pour tout entier naturel $n$ impair.
+
+\item Le calcul de $F_5$ donne $F_5 = 4294967297 = 6700417\times641$. Que dire de $F_5$? Est-ce contradictoire avec la question précédente?
+
+\item Montrer que $F_{n+1} = (F_n - 1)^2 +1$.
+
+\item Montrer que $F_{n+1} - 2$ est divisible par $F_{n}$.
+
+\item Montrer $F_{n+1}$ et $F_{n}$ sont premiers entre eux.
+
+\end{enumerate}
+
+
+
+\end{document}
diff --git a/partiels/131217S1/partiel.tex.bak b/partiels/131217S1/partiel.tex.bak
new file mode 100644
index 0000000..ae07752
--- /dev/null
+++ b/partiels/131217S1/partiel.tex.bak
@@ -0,0 +1,193 @@
+\documentclass[11pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage[amsmath,thmmarks,thref,framed]{ntheorem}
+\usepackage[dvips]{graphics}
+\usepackage{epsfig}
+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{times}
+\usepackage{tabularx}
+\usepackage{textcomp}
+\usepackage{pst-all}
+\usepackage[a4paper]{geometry}
+
+\input{symboles.sty}
+\geometry{hmargin=1cm, tmargin=-0.5cm, bmargin=1.5cm}
+
+\title{
+DUT d'informatique. Partiel de mathématiques discrètes.\\
+Semestre 1 (Décembre 2013). Durée 1h30.}
+
+\date{}
+
+\begin{document}
+\maketitle
+
+\vspace{-3em}
+\noindent Seule une fiche manuscrite de format A5 est autorisée.
+Tous les moyens de communications sont interdits.
+Tous les exercices de ce sujet sont indépendants les uns des autres.
+Dans tout ce qui suit, on considère une algèbre de Boole
+munie des opérateurs classiques ``+'', ``.'',
+``$\overline{\begin{array}{l}~\end{array}}$'' et des variables booléennes
+$a$, $b$, $c$, $d$.
+
+
+\vspace{-1em}
+\section{( pts) Algèbre de Boole }
+On cherche à analyser les codes binaires sur 4 bits.
+
+
+\begin{enumerate}
+\item Dans cette question, un code est correct s'il contient au
+plus deux 1 consécutifs. Par exemple $1010$ est dit correct
+(il ne contient pas de 1 consécutifs) tandis
+$0111$ ne l'est pas (il contient trois 1 consécutifs).
+Le but de cette question est de concevoir la fonction $F$ qui renvoit 1
+si le code de 4 bits $abcd$ est correct et 0 sinon.
+\begin{enumerate}
+\item Ecrire la table de vérité de cet analyseur dans une table de Karnaugh.
+\item Donner l'expression la plus simplifiée de la fonction $F$.
+\end{enumerate}
+
+\begin{enumerate}
+\item Dans cette question, un code est correct
+s'il ne contient pas de 0 consécutifs (par exemple $1011$).
+Le but de cette question est de concevoir la fonction $G$ qui renvoit 1
+si le code de 4 bits $abcd$ est correct et 0 sinon.
+\begin{enumerate}
+\item Ecrire la table de vérité de cet analyseur dans une table de Karnaugh.
+\item Donner l'expression la plus simplifiée de la fonction $G$.
+\end{enumerate}
+\item Dans cette question, code est correct
+s'il vérifie les deux contraintes précédentes.
+Le but de cette question est de concevoir la fonction $H$ qui renvoit 1
+si le code de 4 bits $abcd$ est correct et 0 sinon.
+\begin{enumerate}
+\item Sans utiliser de table de Karnaugh, résoudre algébrique de ce
+problème.
+\item Exprimer $H$ sous la forme d'une somme de trois monomes.
+\item Comment pourrait-on vérifier le résultat sur les tables de Karnaugh
+remplies aux deux premières questions.
+\end{enumerate}
+\end{enumerate}
+
+\vspace{-1em}
+\section{( pts) Relations binaires entre ensembles}
+
+
+\begin{enumerate}
+\item
+Soit $P^*$ l'ensemble des nombres premiers supérieurs à 2. On considère
+la relation $\mathcal{R}$ entre deux éléments de $P^*$ définie par:
+$$
+p \mathcal{R} q \textrm{ si et seulement si }
+\dfrac{p+q}{2} \textrm{ appartient à $P^*$.}
+$$
+Montrer qu'elle n'est pas transitive.
+
+\item On considère
+la relation $\mathcal{S}$ entre deux éléments de $\mathcal{Z}$ définie par:
+$$
+m \mathcal{S} n \textrm{ si et seulement si }
+m^2 + m = n^2 + n.
+$$
+\begin{enumerate}
+\item Montrer que c'est une relation d'équivalence.
+\item Construire la classe de 0, nommée $\dot{0}$.
+\item Pour tout entier relatif $m$, construire la classe de $m$,
+ nommée $\dot{m}$.
+\end{enumerate}
+\end{enumerate}
+
+\vspace{-1em}
+\section{( pts) Démonstration par récurrence}
+On considère la suite $(U_n)_{n \in \N}$ définie pour
+tout $n \in \N$ par $U_n = 10^n.(9n-1)+1$.
+\begin{enumerate}
+\item Calculer $U_0$, $U_1$ et $U_2$.
+\item Montrer que pour tout $n \in \N$ par
+$U_{n+1} = U_n + 9^2.10^n.(n+1)$.
+\item Montrer que $U_n$ est divisible par 81 pour tout $n \in \N$.
+\end{enumerate}
+\end{enumerate}
+
+\vspace{-1em}
+\section{( pts) Plus Grand Commun Diviseur}
+Dans ce qui suit, $n$ est un entier naturel strictement positif.
+On considère trois suites
+$(U_n)_{n \in \N^*}$, $(U_n)_{n \in \N^*}$ et $(U_n)_{n \in \N^*}$
+défines par
+$A_n = n^2 - 2n +2$,
+$B_n = n² +2n +2$
+ et
+$d_n= A_n \land B_n$.
+
+\begin{enumerate}
+\item Calculer $A_1$, $A_2$, $A_3$, $A_4$, $B_1$, $B_2$, $B_3$, $B_4$, $d_1$, $d_2$, $d_3$ et $d_4$. Quelles intuitions avez vous à propos de $d_n$?
+\item Montrer que si un entier naturel
+ $d$ divise à la fois $A_n$ et $n$, alors $d$ divise 2.
+\item Montrer que si un entier naturel
+ $d$ divise à la fois $A_n$ et $B_n$, alors $d$ divise $4n$.
+\item Dans cette question on suppose que $n$ est impair.
+\begin{enumerate}
+\item Montrer que $A_n$ et $B_n$ sont impairs.
+ En déduire que $d_n$ est impair.
+\item Montrer que $d_n$ divise $n$.
+\item En utilisant la question 1, en déduire que $d_n$ divise 2,
+ puis que $A_n$ et $B_n$ sont premiers entre eux.
+\end{enumerate}
+\item On suppose maintenant que $n$ est pair.
+\begin{enumerate}
+\item Montrer que $d_n$ est pair.
+\item Montrer que 4 ne divise pas $A_n$. En déduire la même chose pour $B_n$
+\item Montrer que $d_n$ peut s'écrire sous la forme $d_n = 2.p$,
+ où $p$ est impair.
+\item En utilisant entre autre la question 2., en décuire
+ que $p$ divise $n$.
+\item En utilisant entre autre la question 1., en déduire que $d_n=2$.
+\end{enumerate}
+\end{enumerate}
+
+
+
+\vspace{-1em}
+\section{( pts) Nombres (premiers ou pas) de Fermat}
+Pour $p \in \N$, les nombres de Fermat sont ceux de la forme
+$F_p = 2^{2^p}+1$.
+
+
+\begin{enumerate}
+\item Question préliminaire: montrer que les deux égalités suivantes sont établies:
+\begin{enumerate}
+\item $x^n- 1 = (x-1)(x^{n-1}+x^{n-2}+\ldots+ x + 1)$ pour tout entier naturel $n$ strictement positif.
+\end{enumerate}
+
+\item
+Montrer que si $n$ n'est pas une puissance de 2, alors
+$2^n+1$ n'est pas un nombre premier.
+On pourra se servir de l'égalité
+$$
+x^n+ 1 = (x+1)(x^{n-1}-x^{n-2}+\ldots - x + 1)
+$$ qui est établie
+pour tout entier naturel $n$ impair.
+
+\item Le calcul de $F_5$ donne $F_5 = 4294967297 = 6700417\times641$. Que dire de $F_5$? Est-ce contradictoire avec la question précédente?
+
+\item Montrer que, $F_{n+1} = (F_n - 1)^2 +1$.
+
+\item Montrer que $F_{n+1} - 2$ est divisible par $F_{n}$.
+
+\item Montrer $F_{n+1}$ et $F_{n}$ sont premiers entre eux.
+
+\end{enumerate}
+
+
+
+\end{document}
diff --git a/partiels/131217S1/partiel.tex~ b/partiels/131217S1/partiel.tex~
new file mode 100644
index 0000000..41af89a
--- /dev/null
+++ b/partiels/131217S1/partiel.tex~
@@ -0,0 +1,193 @@
+\documentclass[11pt,a4paper,french]{article}
+\usepackage[francais]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{a4}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{framed}
+\usepackage[amsmath,thmmarks,thref,framed]{ntheorem}
+\usepackage[dvips]{graphics}
+\usepackage{epsfig}
+\usepackage{calc}
+\usepackage{tabls}
+\usepackage{times}
+\usepackage{tabularx}
+\usepackage{textcomp}
+\usepackage{pst-all}
+\usepackage[a4paper]{geometry}
+
+\input{symboles.sty}
+\geometry{hmargin=1cm, tmargin=-0.5cm, bmargin=1.5cm}
+
+\title{
+DUT d'informatique. Partiel de mathématiques discrètes.\\
+Semestre 1 (Décembre 2013). Durée 1h30.}
+
+\date{}
+
+\begin{document}
+\maketitle
+
+\vspace{-3em}
+\noindent Seule une fiche manuscrite de format A5 est autorisée.
+Tous les moyens de communications sont interdits.
+Tous les exercices de ce sujet sont indépendants les uns des autres.
+Dans tout ce qui suit, on considère une algèbre de Boole
+munie des opérateurs classiques ``+'', ``.'',
+``$\overline{\begin{array}{l}~\end{array}}$'' et des variables booléennes
+$a$, $b$, $c$, $d$.
+
+
+\vspace{-1em}
+\section{( pts) Algèbre de Boole }
+On cherche à analyser les codes binaires sur 4 bits.
+
+
+\begin{enumerate}
+\item Dans cette question, un code est correct s'il contient au
+plus deux 1 consécutifs. Par exemple $1010$ est dit correct
+(il ne contient pas de 1 consécutifs) tandis
+$0111$ ne l'est pas (il contient trois 1 consécutifs).
+Le but de cette question est de concevoir la fonction $F$ qui renvoie 1
+si le code de 4 bits $abcd$ est correct et 0 sinon.
+\begin{enumerate}
+\item Ãcrire la table de vérité de cet analyseur dans une table de Karnaugh.
+\item Donner l'expression la plus simplifiée de la fonction $F$.
+\end{enumerate}
+
+\begin{enumerate}
+\item Dans cette question, un code est correct
+s'il ne contient pas de 0 consécutifs (par exemple $1011$).
+Le but de cette question est de concevoir la fonction $G$ qui renvoie 1
+si le code de 4 bits $abcd$ est correct et 0 sinon.
+\begin{enumerate}
+\item Ãcrire la table de vérité de cet analyseur dans une table de Karnaugh.
+\item Donner l'expression la plus simplifiée de la fonction $G$.
+\end{enumerate}
+\item Dans cette question, code est correct
+s'il vérifie les deux contraintes précédentes.
+Le but de cette question est de concevoir la fonction $H$ qui renvoie 1
+si le code de 4 bits $abcd$ est correct et 0 sinon.
+\begin{enumerate}
+\item Sans utiliser de table de Karnaugh, résoudre algébrique de ce
+problème.
+\item Exprimer $H$ sous la forme d'une somme de trois monômes.
+\item Comment pourrait-on vérifier le résultat sur les tables de Karnaugh
+remplies aux deux premières questions.
+\end{enumerate}
+\end{enumerate}
+
+\vspace{-1em}
+\section{( pts) Relations binaires entre ensembles}
+
+
+\begin{enumerate}
+\item
+Soit $P^*$ l'ensemble des nombres premiers supérieurs à 2. On considère
+la relation $\mathcal{R}$ entre deux éléments de $P^*$ définie par:
+$$
+p \mathcal{R} q \textrm{ si et seulement si }
+\dfrac{p+q}{2} \textrm{ appartient à $P^*$.}
+$$
+Montrer qu'elle n'est pas transitive.
+
+\item On considère
+la relation $\mathcal{S}$ entre deux éléments de $\mathcal{Z}$ définie par:
+$$
+m \mathcal{S} n \textrm{ si et seulement si }
+m^2 + m = n^2 + n.
+$$
+\begin{enumerate}
+\item Montrer que c'est une relation d'équivalence.
+\item Construire la classe de 0, nommée $\dot{0}$.
+\item Pour tout entier relatif $m$, construire la classe de $m$,
+ nommée $\dot{m}$.
+\end{enumerate}
+\end{enumerate}
+
+\vspace{-1em}
+\section{( pts) Démonstration par récurrence}
+On considère la suite $(U_n)_{n \in \N}$ définie pour
+tout $n \in \N$ par $U_n = 10^n.(9n-1)+1$.
+\begin{enumerate}
+\item Calculer $U_0$, $U_1$ et $U_2$.
+\item Montrer que pour tout $n \in \N$ par
+$U_{n+1} = U_n + 9^2.10^n.(n+1)$.
+\item Montrer que $U_n$ est divisible par 81 pour tout $n \in \N$.
+\end{enumerate}
+\end{enumerate}
+
+\vspace{-1em}
+\section{( pts) Plus Grand Commun Diviseur}
+Dans ce qui suit, $n$ est un entier naturel strictement positif.
+On considère trois suites
+$(U_n)_{n \in \N^*}$, $(U_n)_{n \in \N^*}$ et $(U_n)_{n \in \N^*}$
+définies par
+$A_n = n^2 - 2n +2$,
+$B_n = n² +2n +2$
+ et
+$d_n= A_n \land B_n$.
+
+\begin{enumerate}
+\item Calculer $A_1$, $A_2$, $A_3$, $A_4$, $B_1$, $B_2$, $B_3$, $B_4$, $d_1$, $d_2$, $d_3$ et $d_4$. Quelles intuitions avez vous à propos de $d_n$?
+\item Montrer que si un entier naturel
+ $d$ divise à la fois $A_n$ et $n$, alors $d$ divise 2.
+\item Montrer que si un entier naturel
+ $d$ divise à la fois $A_n$ et $B_n$, alors $d$ divise $4n$.
+\item Dans cette question on suppose que $n$ est impair.
+\begin{enumerate}
+\item Montrer que $A_n$ et $B_n$ sont impairs.
+ En déduire que $d_n$ est impair.
+\item Montrer que $d_n$ divise $n$.
+\item En utilisant la question 1, en déduire que $d_n$ divise 2,
+ puis que $A_n$ et $B_n$ sont premiers entre eux.
+\end{enumerate}
+\item On suppose maintenant que $n$ est pair.
+\begin{enumerate}
+\item Montrer que $d_n$ est pair.
+\item Montrer que 4 ne divise pas $A_n$. En déduire la même chose pour $B_n$
+\item Montrer que $d_n$ peut s'écrire sous la forme $d_n = 2.p$,
+ où $p$ est impair.
+\item En utilisant entre autre la question 2., en déduire
+ que $p$ divise $n$.
+\item En utilisant entre autre la question 1., en déduire que $d_n=2$.
+\end{enumerate}
+\end{enumerate}
+
+
+
+\vspace{-1em}
+\section{( pts) Nombres (premiers ou pas) de Fermat}
+Pour $p \in \N$, les nombres de Fermat sont ceux de la forme
+$F_p = 2^{2^p}+1$.
+
+
+\begin{enumerate}
+\item Question préliminaire: montrer que les deux égalités suivantes sont établies:
+\begin{enumerate}
+\item $x^n- 1 = (x-1)(x^{n-1}+x^{n-2}+\ldots+ x + 1)$ pour tout entier naturel $n$ strictement positif.
+\end{enumerate}
+
+\item
+Montrer que si $n$ n'est pas une puissance de 2, alors
+$2^n+1$ n'est pas un nombre premier.
+On pourra se servir de l'égalité
+$$
+x^n+ 1 = (x+1)(x^{n-1}-x^{n-2}+\ldots - x + 1)
+$$ qui est établie
+pour tout entier naturel $n$ impair.
+
+\item Le calcul de $F_5$ donne $F_5 = 4294967297 = 6700417\times641$. Que dire de $F_5$? Est-ce contradictoire avec la question précédente?
+
+\item Montrer que, $F_{n+1} = (F_n - 1)^2 +1$.
+
+\item Montrer que $F_{n+1} - 2$ est divisible par $F_{n}$.
+
+\item Montrer $F_{n+1}$ et $F_{n}$ sont premiers entre eux.
+
+\end{enumerate}
+
+
+
+\end{document}
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