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% French language support
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% --- Basic Settings ---
% Language variables for French exams
\def\professor{MOSAID}  % Should be MOSAID not {{MOSAID}}
\def\classname{1BAC.SM}
\def\examtitle{Exercises devoirs Arithmétiques dans $\mathbb{Z}$ }
\def\schoolname{Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{visit ~~\wsite~~ for more!}}
\def\province{Direction provinciale\newline de Beni Mellal}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}


% Exam-specific settings and custom definitions
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% Custom theorem environments (uncomment if needed)
% \newtheorem{theorem}{Theorem}
% \newtheorem{lemma}{Lemma}
% \newtheorem{corollary}{Corollary}

% Custom exercise environments (uncomment if needed)
% \newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
% \newenvironment{remark}{\begin{proof}[Remark]}{\end{proof}}

% Custom commands for this exam (add your own below)
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% \newcommand{\points}[1]{\hfill(\mbox{\#1 points})}

% Additional packages (uncomment if needed)
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% \usepackage{circuitikz} % For circuit diagrams

% Page layout adjustments
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% Custom colors for this exam
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% ----------------------------------------------------------------------
% ADD YOUR CUSTOM DEFINITIONS BELOW THIS LINE
% ----------------------------------------------------------------------

% Example:
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\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}


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\begin{document}
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% Two-column layout
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\begin{multicols}{2}[\raggedcolumns]
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% Header for first column in landscape mode
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% Exercise 1 (originally 1)
\printexo{1}{}{
On se propose de déterminer l'ensemble \((E)\) des couples \((x; y)\) d'entiers naturels tel que
~\(
2 \cdot 3^y = 1 + 5^x.
\)~\\
Soit \((x; y)\) un élément de \((E)\) tel que \(x \geq 1\) et \(y \geq 1\).
\begin{enumerate}
\item Montrer que
~\(
1 + (-1)^x \equiv 0 \pmod{3}.
\)~

\item En déduire que
~\(
5^x \equiv 5 \pmod{8}.
\)~

\item
\begin{enumerate}
\item Déterminer, suivant les valeurs de l'entier naturel \(n\), les restes modulo 8 de \(3^n\).
\item En déduire que \(y\) est impair.
\end{enumerate}

\item On suppose dans cette question que \(y \equiv 0 \pmod{3}\).
\begin{enumerate}
\item Justifier que
~\(
3^y \equiv -1 \pmod{7}.
\)~
\item Déterminer, suivant les valeurs de l'entier naturel \(m\), les restes modulo 7 de \(5^m\).
\item Montrer alors que
~\(
x \equiv 2 \pmod{6}.
\)~
\item Conclure.
\end{enumerate}

\item On suppose dans cette question que \(y \equiv 1 \pmod{3}\).
\begin{enumerate}
\item Justifier que
~\(
3^y \equiv 3 \pmod{7}.
\)~
\item En déduire que
~\(
5^x \equiv 5 \pmod{7}.
\)~
\item Déterminer alors les restes modulo 9 de \(5^x\) et \(3^y\).
\item Déterminer alors \(x\) et \(y\).
\end{enumerate}

\item On suppose dans cette question que \(y \equiv 2 \pmod{3}\).
\begin{enumerate}
\item Déterminer les restes possibles modulo 13 de \(3^y\).
\item En déduire le reste modulo 13 de \(5^x\).
\item Conclure.
\end{enumerate}

\item Déterminer l'ensemble \((E)\).
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
\textbf{Les parties A et B sont indépendantes.}\\
\textbf{Partie A}:
Soit dans \(\mathbb{N} \times \mathbb{N}\) l'équation \(E_1 : 3^x - y^2 = 8\).
\begin{enumerate}
\item Vérifier que le couple \((2,1)\) est une solution de \(E_1\).\\
Soit le couple \((x,y)\) solution de \(E_1\).

\item
\begin{enumerate}
    \item Montrer que \(y\) est impair.
    \item En déduire que \(y^2 \equiv 1 \pmod{8}\).
    \item Montrer que si \(x\) est impair alors \(3^x \equiv 3 \pmod{8}\).
    \item En déduire que \(x\) ne peut être qu'un nombre pair.
\end{enumerate}

\item Soit \(k\) un entier naturel tel que \(x = 2k\).
\begin{enumerate}
    \item Montrer que \((3^k - y)(3^k + y) = 8\).
    \item En déduire que \((2,1)\) est l'unique solution de \(E_1\).
\end{enumerate}
\end{enumerate}
\textbf{Partie B}:
Soit dans \(\mathbb{N}\) l'équation \(E_2 : 12^x \equiv 2 \pmod{23}\).

\begin{enumerate}
\item
\begin{enumerate}
    \item Vérifier que \(12^{22} \equiv 1 \pmod{23}\).
    \item En déduire que \(12^{21} \equiv 2 \pmod{23}\).
\end{enumerate}

\item Soit \(x\) une solution de \(E_2\) et \(d\) le PGCD des entiers \(x+1\) et \(22\).
\begin{enumerate}
    \item Montrer que \(12^{x+1} \equiv 1 \pmod{23}\).
    \item Montrer que \(12^d \equiv 1 \pmod{23}\).
    \item En déduire que \(d = 11\) ou \(d = 22\).
    \item Montrer de ce qui précède que \(x \equiv 10 \pmod{11}\).
\end{enumerate}

\item Résoudre dans \(\mathbb{N}\) l'équation \(E_2\).
\end{enumerate}
}




\end{multicols}


\end{document}