-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{8}}{}}{31}{Exo.8}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{9}}{Diagrammes de transitivité}}{31}{Exo.9}}
-\@writefile{toc}{\contentsline {section}{\numberline {III}Relations d'\IeC {\'e}quivalence}{32}{section.4.3}}
-\@writefile{thm}{\contentsline {Def}{{Définition}{4.{6}}{Relation symétrique}}{32}{Def.6}}
-\@writefile{thm}{\contentsline {Def}{{Définition}{4.{7}}{Relation d'équivalence}}{32}{Def.7}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{10}}{}}{32}{Exo.10}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{11}}{Relation de congruence modulo $n$ dans $\Z $}}{32}{Exo.11}}
-\global\def\markxxiiExi{\ensuremath {}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{12}}{}}{32}{Exo.12}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{32}{subsection.4.3.1}}
-\@writefile{thm}{\contentsline {Def}{{Définition}{4.{8}}{Classe d'équivalence}}{32}{Def.8}}
-\@writefile{thm}{\contentsline {Notation}{{Notation}{4.{1}}{}}{32}{Notation.1}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{13}}{}}{32}{Exo.13}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{14}}{}}{32}{Exo.14}}
-\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{1}}{}}{33}{Th.1}}
-\@writefile{thm}{\contentsline {Rem}{{Remarque}{4.{2}}{}}{33}{Rem.2}}
-\@writefile{thm}{\contentsline {Pre}{{Preuve}{1}{}}{33}{Pre.1}}
-\@writefile{thm}{\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{33}{Def.9}}
-\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{2}}{}}{33}{Th.2}}
-\@writefile{thm}{\contentsline {Pre}{{Preuve}{2}{}}{33}{Pre.2}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{15}}{}}{33}{Exo.15}}
-\global\def\markxxiiiEx{\ensuremath {}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{16}}{}}{33}{Exo.16}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{17}}{Une relation d'équivalence}}{33}{Exo.17}}
-\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{18}}{}}{33}{Exo.18}}
-\@writefile{toc}{\contentsline {subsection}{\numberline {III.2}Ensemble-quotient}{34}{subsection.4.3.2}}
-\@writefile{thm}{\contentsline {Def}{{Définition}{4.{10}}{Ensemble-quotient}}{34}{Def.10}}
-\@writefile{thm}{\contentsline {Notation}{{Notation}{4.{2}}{}}{34}{Notation.2}}
-\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{19}}{Congruence modulo 4}}{34}{Exo.19}}
-\@writefile{toc}{\contentsline {part}{III\hspace {1em}Annexes}{35}{part.3}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{10}}{}}{25}{Exo.4.10}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {III.1}Classes d'\IeC {\'e}quivalence}{26}{subsection.4.3.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{8}}{Classe d'équivalence}}{26}{Def.4.8}}
+\@writefile{thm}{\contentsline {Notation}{{Notation}{4.{1}}{}}{26}{Notation.4.1}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{11}}{}}{26}{Exo.4.11}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{12}}{}}{26}{Exo.4.12}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{1}}{}}{26}{Th.4.1}}
+\@writefile{thm}{\contentsline {Pre}{{Preuve}{1}{}}{26}{Pre.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{4.{9}}{Partition d'un ensemble}}{26}{Def.4.9}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{4.{2}}{}}{26}{Th.4.2}}
+\@writefile{thm}{\contentsline {Pre}{{Preuve}{2}{}}{26}{Pre.2}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{4.{13}}{}}{26}{Exo.4.13}}
+\global\def\markxxiEx{\ensuremath {}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{14}}{}}{26}{Exo.4.14}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{4.{15}}{}}{26}{Exo.4.15}}
+\@writefile{toc}{\contentsline {part}{III\hspace {1em}Arithm\IeC {\'e}tique}{27}{part.3}}
+\@writefile{toc}{\contentsline {chapter}{\numberline {5}Ensembles de nombres entiers}{28}{chapter.5}}
+\@writefile{lof}{\addvspace {10\p@ }}
+\@writefile{lot}{\addvspace {10\p@ }}
+\@writefile{toc}{\contentsline {section}{\numberline {I}Principe de r\IeC {\'e}currence }{28}{section.5.1}}
+\newlabel{itm:2}{{2}{28}{Principe de récurrence}{Item.147}{}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{1}}{}}{28}{Exo.5.1}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{2}}{}}{28}{Exo.5.2}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{3}}{}}{28}{Exo.5.3}}
+\@writefile{toc}{\contentsline {section}{\numberline {II}Nombres premiers}{28}{section.5.2}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{1}}{Multiple, diviseur}}{28}{Def.5.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{2}}{Nombre premier}}{28}{Def.5.2}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{1}}{}}{28}{Rem.5.1}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{3}}{Décomposition en facteurs premiers}}{28}{Def.5.3}}
+\global\def\markxxxiDefi{\ensuremath {}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{1}}{}}{28}{Th.5.1}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{4}}{}}{29}{Exo.5.4}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{5}}{}}{29}{Exo.5.5}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{2}}{}}{29}{Th.5.2}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{6}}{Nombres premiers en quantité infinie}}{29}{Exo.5.6}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{4}}{PGCD, PPCM}}{29}{Def.5.4}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{5}}{Nombres premiers entre eux}}{29}{Def.5.5}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{7}}{Nombres de Fermat}}{29}{Exo.5.7}}
+\@writefile{toc}{\contentsline {section}{\numberline {III}Division euclidienne dans ${\mathbb Z}$ et applications}{29}{section.5.3}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{3}}{}}{29}{Th.5.3}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{6}}{Division euclidienne}}{29}{Def.5.6}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{8}}{}}{29}{Exo.5.8}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{9}}{}}{29}{Exo.5.9}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{10}}{}}{29}{Exo.5.10}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{11}}{Numéro Sécurité Sociale (\url {www.bibmath.net})}}{30}{Exo.5.11}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{12}}{}}{30}{Exo.5.12}}
+\@writefile{toc}{\contentsline {section}{\numberline {IV}Algorithmes d'Euclide}{30}{section.5.4}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {IV.1}L'algorithme initial}{30}{subsection.5.4.1}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{2}}{}}{31}{Rem.5.2}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{13}}{}}{31}{Exo.5.13}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{14}}{}}{31}{Exo.5.14}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{15}}{}}{31}{Exo.5.15}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{4}}{Théorème de Bézout}}{31}{Th.5.4}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\alpha $}}{}}{31}{Proof.5.1}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{3}}{}}{31}{Rem.5.3}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\beta $}}{}}{31}{Proof.5.2}}
+\global\def\markxRemi{\ensuremath {}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{16}}{Application de l'algorithme d'Euclide et de Bézout}}{31}{Exo.5.16}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {IV.2}Algorithme d'Euclide g\IeC {\'e}n\IeC {\'e}ralis\IeC {\'e}}{32}{subsection.5.4.2}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {IV.3}L'algorithme.}{32}{subsection.5.4.3}}
+\@writefile{thm}{\contentsline {Pre}{{Preuve}{3}{}}{32}{Pre.3}}
+\global\def\markiiiPrei{$\dagger $}
+\@writefile{toc}{\contentsline {subsection}{\numberline {IV.4}Exemple.}{32}{subsection.5.4.4}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{17}}{}}{32}{Exo.5.17}}
+\global\def\markxxivEx{\ensuremath {}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{4}}{}}{32}{Rem.5.4}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{18}}{}}{32}{Exo.5.18}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{5}}{Théorème de Gauss}}{32}{Th.5.5}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{19}}{}}{33}{Exo.5.19}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{20}}{}}{33}{Exo.5.20}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{21}}{}}{33}{Exo.5.21}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{22}}{}}{33}{Exo.5.22}}
+\@writefile{toc}{\contentsline {section}{\numberline {V}Arithm\IeC {\'e}tique modulo $n$}{33}{section.5.5}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{7}}{Congruence modulo $n$}}{33}{Def.5.7}}
+\global\def\markxxxvDef{\ensuremath {}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{23}}{}}{33}{Exo.5.23}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{6}}{}}{33}{Th.5.6}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\gamma $}}{}}{33}{Proof.5.3}}
+\global\def\markviiiProofi{\rule {1ex}{1ex}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{24}}{}}{34}{Exo.5.24}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{7}}{}}{34}{Th.5.7}}
+\@writefile{thm}{\contentsline {Notation}{{Notation}{5.{1}}{}}{34}{Notation.5.1}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{25}}{}}{34}{Exo.5.25}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{8}}{}}{34}{Def.5.8}}
+\global\def\markxxxviDefi{\ensuremath {}}
+\@writefile{thm}{\contentsline {Th}{{Propriété}{5.{8}}{}}{34}{Th.5.8}}
+\@writefile{thm}{\contentsline {Proof}{{Preuve}{5.{$\delta $}}{}}{34}{Proof.5.4}}
+\global\def\markixProofii{\rule {1ex}{1ex}}
+\@writefile{thm}{\contentsline {Def}{{Définition}{5.{9}}{}}{34}{Def.5.9}}
+\@writefile{thm}{\contentsline {Ex}{{Exemple}{5.{26}}{}}{34}{Exo.5.26}}
+\global\def\markxxviiExi{\ensuremath {}}
+\@writefile{thm}{\contentsline {Rem}{{Remarque}{5.{5}}{}}{35}{Rem.5.5}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{27}}{}}{35}{Exo.5.27}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{28}}{Systèmes de congruences}}{35}{Exo.5.28}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{29}}{}}{35}{Exo.5.29}}
+\@writefile{thm}{\contentsline {Exo}{{Exercice}{5.{30}}{}}{35}{Exo.5.30}}
+\@writefile{toc}{\contentsline {part}{IV\hspace {1em}Annexes}{36}{part.4}}