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\begin{document}

\begin{center}
    \small Les 6 exercices d'arithmétique ci-dessous peuvent être réalisés facilement avec une calculatrice. Mais maîtrisez-vous les automatismes nécessaires pour vous en passer ? Division euclidienne, multiples, décomposition en facteurs premiers, PPCM, PGCD, fractions irréductibles et nombres premiers : un tour d'horizon complet du programme de cycle 4, utile pour réviser le DNB.
\end{center}
\vspace{0.5em}
\sloppy


{\color{blue}

« Les 6 exercices suivants peuvent être réalisés facilement avec une calculatrice. Mais maîtrisez-vous les automatismes nécessaires pour vous en passer ? »

}

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 1}}\par

{\color{green!60!black}
    Encadre 348 et 187 par deux multiples consécutifs de 8.
    }

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 2}}\par

{\color{green!60!black}
        Quel est le plus grand multiple de 18 inférieur à 376 ?
        }

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 3}}\par

{\color{green!60!black}
        Quel est le plus petit multiple de 23 supérieur à 421 ?
        }

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 4}}\par

{\color{green!60!black}
\begin{enumerate}
\item Décompose 4256 et 7047 en produit de facteurs premiers.
 \item Déduis leur PPCM (Plus petit multiple commun)
 \item Déduis en leur PGCD (Plus grand diviseur commun)
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{4256}{7047}\)
\end{enumerate}
}

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 5}}\par

{\color{green!60!black}
\begin{enumerate}
\item Donne les listes ordonnées des diviseurs de  24 et 874.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 24 et 874.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{24}{874}\)
\end{enumerate}
}

"
\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 6}}\par

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 877;  10362;  925; 481 }

 \newpage
\section*{\textcolor{red}{Correction}}

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 1}}\par

{\color{green!60!black}
        Encadre 348 et 187 par deux multiples consécutifs de 8.
        }

On effectue la division euclidienne de 348 par 8 :

 
\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[gray] (1.540,1.430) -- (1.540,1.320);
  \draw[black] (1.980,1.705) -- (1.980,0.220);
  \draw[black] (1.980,1.430) -- (2.860,1.430);
  \draw[black] (1.045,1.265) -- (1.375,1.265);
  \draw[black] (1.265,0.825) -- (1.595,0.825);
  \draw[blue] (0.990,1.705) .. controls (1.137,1.778) and (1.283,1.778) .. (1.430,1.705);
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.540) {3};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {4};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.540) {8};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {8};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {4};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 348 = 8 \(\times\) 43 + 4 et 4 \(<\) 8
 \item[] 348 = 344 + 4
 \item[] {\color{red}
donc 344 \(<\) 348 \(<\) 352 (344 + 8)}
\end{itemize}

De même:

On effectue la division euclidienne de 187 par 8 :

 
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  \draw[gray] (1.540,1.430) -- (1.540,1.320);
  \draw[black] (1.980,1.705) -- (1.980,0.220);
  \draw[black] (1.980,1.430) -- (2.860,1.430);
  \draw[black] (1.045,1.265) -- (1.375,1.265);
  \draw[black] (1.265,0.825) -- (1.595,0.825);
  \draw[blue] (0.990,1.705) .. controls (1.137,1.778) and (1.283,1.778) .. (1.430,1.705);
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.540) {1};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {3};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 187 = 8 \(\times\) 23 + 3 et 3 \(<\) 8
 \item[] 187 = 184 + 3
 \item[] {\color{red}
donc 184 \(<\) 187 \(<\) 192 (184 + 8)}
\end{itemize}

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 2}}\par

{\color{green!60!black}
           Quel est le plus grand multiple de 18 inférieur à 376 ?
           }

On effectue la division euclidienne de 376 par 18 :

 
\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[gray] (1.540,1.430) -- (1.540,1.320);
  \draw[black] (1.980,1.705) -- (1.980,0.220);
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  \draw[black] (1.265,0.825) -- (1.595,0.825);
  \draw[blue] (0.990,1.705) .. controls (1.137,1.778) and (1.283,1.778) .. (1.430,1.705);
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  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {18};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.210) {2};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.420,1.210) {0};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {6};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.320) {3};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.605) {1};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 376 = 18 \(\times\) 20 + 16 et 16 \(<\) 18
\item[] {\color{red} Le plus grand multiple de 18 inférieur à 376 est donc 18 \(\times\) 20 = 360.}
\end{itemize}

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 3}}\par

{\color{green!60!black}
            Quel est le plus petit multiple de 23 supérieur à 421 ?
           }

On effectue la division euclidienne de 421 par 23 :

 
\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[gray] (1.540,1.430) -- (1.540,1.320);
  \draw[black] (1.980,1.705) -- (1.980,0.220);
  \draw[black] (1.980,1.430) -- (2.860,1.430);
  \draw[black] (1.045,1.265) -- (1.375,1.265);
  \draw[black] (1.045,0.825) -- (1.595,0.825);
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  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.540) {4};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {2};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.540) {1};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {23};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.210) {1};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.420,1.210) {8};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {3};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.320) {2};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.100) {1};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {9};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.100) {1};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.880) {4};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.880) {8};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,0.880) {1};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {7};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 421 = 23 \(\times\) 18 + 7 et 7 \(<\) 23
\item[] {\color{red} Le plus petit multiple de 23 inférieur à 421 est donc 23 \(\times\) 19 = 437.}
\end{itemize}

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 4}}\par

{\color{green!60!black}
    Décomposition de  4256 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
4256 & 2\\
2128 & 2\\
1064 & 2\\
532 & 2\\
266 & 2\\
133 & 7\\
19 & 19\\
1 & \\
\end{tabular}\qquad 4256 = 2\textsuperscript{5} $\times$ 7 $\times$ 19
\end{center}}

    Décomposition de  7047 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
7047 & 3\\
2349 & 3\\
783 & 3\\
261 & 3\\
87 & 3\\
29 & 29\\
1 & \\
\end{tabular}\qquad 7047 = 3\textsuperscript{5} $\times$ 29
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 4256 = 2\textsuperscript{5} \(\times\) 7 \(\times\) 19}
\par\addvspace{.55em}
{\color{green!60!black} 7047 = 3\textsuperscript{5} \(\times\) 29}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(4256;7047) = 2\textsuperscript{5} \(\times\) 3\textsuperscript{5} \(\times\) 7 \(\times\) 19 \(\times\) 29 = 29992032}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(4256,7047) =  1}}
 \item {\color{red} Simplification de la fraction :
\par\addvspace{.55em}
 {\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
4256 et 7047 sont premiers entre eux.
\par\addvspace{.55em}

         Donc la fraction \(\dfrac{4256}{7047}\) est irréductible.}}
\end{enumerate}
}

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 5}}\par

{\color{green!60!black}
\begin{enumerate}
\item {\color{red} Les diviseurs :

{\color{green!60!black} 24 : \{ 1; 2; 3; 4; 6; 8; 12; 24 \}
\par\addvspace{.55em}

874 : \{ 1; 2; 19; 23; 38; 46; 437; 874 \}}

}
 \item {\color{red} Les diviseurs communs de 24 et 874 sont :

{\color{green!60!black}
\{ 1; 2 \}}

 }
\item {\color{red} Le plus grand diviseur commun de 24 et 874 est :

{\color{green!60!black}
PGCD(24;874) = 2}

}
 \item {\color{red} Simplification de la fraction :

{\color{green!60!black} Si on simplifie une fraction par
                 le plus grand diviseur commun à son numérateur
                 et son dénominateur, la fraction obtenue est irréductible.}

 {\color{green!60!black} d'où \(\dfrac{24}{874}\) = \(\dfrac{24:2}{874:2}\) = \(\dfrac{12}{437}\) } }
\end{enumerate}
}

\needspace{3\baselineskip}\par\smallskip\noindent{\color{red}\textbf{Exercice 6}}\par

\begin{enumerate}
\item {\color{red}
        877 est-il premier ?

{\color{green!60!black}  877 = 2 \(\times\) 438 + 1 877 = 3 \(\times\) 292 + 1 877 = 5 \(\times\) 175 + 2 877 = 7 \(\times\) 125 + 2 877 = 11 \(\times\) 79 + 8 877 = 13 \(\times\) 67 + 6 877 = 17 \(\times\) 51 + 10 877 = 19 \(\times\) 46 + 3 877 = 23 \(\times\) 38 + 3 877 = 29 \(\times\) 30 + 7 877 = 31 \(\times\) 28 + 9
\par\addvspace{.55em}

            877 n'a pas d'autres diviseurs que 1 et 877  donc \emph{ 877 est un nombre premier.} }

}
\item {\color{red}
        10362 est-il premier ?

{\color{green!60!black}

        10362 est pair donc \emph{10362 n'est pas un nombre premier}.}

}

\item {\color{red}
        925 est-il premier ?

{\color{green!60!black}

        925 se termine par un 5, c'est un multiple de 5 donc \emph{925 n'est pas un nombre premier}.}

}

\item {\color{red}
        481 est-il premier ?

{\color{green!60!black}  481 = 2 \(\times\) 240 + 1 481 = 3 \(\times\) 160 + 1 481 = 5 \(\times\) 96 + 1 481 = 7 \(\times\) 68 + 5 481 = 11 \(\times\) 43 + 8 481 = 13 \(\times\) 37 + 0
\par\addvspace{.55em}

            481 est divisible par 13 donc\emph{  481 n'est pas un nombre premier}.
            }

}
\end{enumerate}
\end{document}
