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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 72 et 446 par deux multiples consécutifs de 15.
    }

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

{\color{green!60!black}
        Quel est le plus grand multiple de 23 inférieur à 550 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 22 supérieur à 371 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 11160 et 6561 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}
\(\dfrac{11160}{6561}\)
\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  11600 et 11907.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 11600 et 11907.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{11600}{11907}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 3427;  617;  15978; 1655 }

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

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

{\color{green!60!black}
        Encadre 72 et 446 par deux multiples consécutifs de 15.
        }

On effectue la division euclidienne de 72 par 15 :

 
\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[black] (1.760,1.265) -- (1.760,0.220);
  \draw[black] (1.760,0.990) -- (2.420,0.990);
  \draw[black] (1.045,0.825) -- (1.375,0.825);
  \draw[blue] (0.990,1.265) .. controls (1.137,1.338) and (1.283,1.338) .. (1.430,1.265);
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.100) {7};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {2};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.980,1.100) {15};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.980,0.770) {4};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,0.605) {1};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 72 = 15 \(\times\) 4 + 12 et 12 \(<\) 15
 \item[] 72 = 60 + 12
 \item[] {\color{red}
donc 60 \(<\) 72 \(<\) 75 (60 + 15)}
\end{itemize}

De même:

On effectue la division euclidienne de 446 par 15 :

 
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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] (1.045,0.825) -- (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[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.420,1.210) {9};
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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[] 446 = 15 \(\times\) 29 + 11 et 11 \(<\) 15
 \item[] 446 = 435 + 11
 \item[] {\color{red}
donc 435 \(<\) 446 \(<\) 450 (435 + 15)}
\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 23 inférieur à 550 ?
           }

On effectue la division euclidienne de 550 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] (1.265,0.825) -- (1.595,0.825);
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.605) {2};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 550 = 23 \(\times\) 23 + 21 et 21 \(<\) 23
\item[] {\color{red} Le plus grand multiple de 23 inférieur à 550 est donc 23 \(\times\) 23 = 529.}
\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 22 supérieur à 371 ?
           }

On effectue la division euclidienne de 371 par 22 :

 
\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) {3};
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  \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) {22};
  \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) {6};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {2};
  \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) {5};
  \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) {2};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.880) {3};
  \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) {9};
  \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[] 371 = 22 \(\times\) 16 + 19 et 19 \(<\) 22
\item[] {\color{red} Le plus petit multiple de 22 inférieur à 371 est donc 22 \(\times\) 17 = 374.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  11160 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
11160 & 2\\
5580 & 2\\
2790 & 2\\
1395 & 3\\
465 & 3\\
155 & 5\\
31 & 31\\
1 & \\
\end{tabular}\qquad 11160 = 2\textsuperscript{3} $\times$ 3\textsuperscript{2} $\times$ 5 $\times$ 31
\end{center}}

    Décomposition de  6561 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
6561 & 3\\
2187 & 3\\
729 & 3\\
243 & 3\\
81 & 3\\
27 & 3\\
9 & 3\\
3 & 3\\
1 & \\
\end{tabular}\qquad 6561 = 3\textsuperscript{8}
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 11160 = 2\textsuperscript{3} \(\times\) 3\textsuperscript{2} \(\times\) 5 \(\times\) 31}
\par\addvspace{.55em}
{\color{green!60!black} 6561 = 3\textsuperscript{8}}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(11160;6561) = 2\textsuperscript{3} \(\times\) 3\textsuperscript{8} \(\times\) 5 \(\times\) 31 = 8135640}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(11160;6561) = 3\textsuperscript{2} = 9}}
\item {\color{red} Simplification de la fraction :
\par\addvspace{.55em}

{\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 \(\dfrac{11160}{6561}\) = \(\dfrac{11160:9}{6561:9}\) = \(\dfrac{1240}{729}\) } }
\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} 11600 : \{ 1; 2; 4; 5; 8; 10; 16; 20; 25; 29; 40; 50; 58; 80; 100; 116; 145; 200; 232; 290; 400; 464; 580; 725; 1160; 1450; 2320; 2900; 5800; 11600 \}
\par\addvspace{.55em}

11907 : \{ 1; 3; 7; 9; 21; 27; 49; 63; 81; 147; 189; 243; 441; 567; 1323; 1701; 3969; 11907 \}}

}
 \item {\color{red} Les diviseurs communs de 11600 et 11907 sont :

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

 }
\item {\color{red} Le plus grand diviseur commun de 11600 et 11907 est :

{\color{green!60!black}
PGCD(11600;11907) = 1}

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

{\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
11600 et 11907 sont premiers entre eux.}

 {\color{green!60!black} Donc la fraction \(\dfrac{11600}{11907}\) est irréductible.}}
\end{enumerate}
}

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

\begin{enumerate}
\item {\color{red}
        3427 est-il premier ?

{\color{green!60!black}  3427 = 2 \(\times\) 1713 + 1 3427 = 3 \(\times\) 1142 + 1 3427 = 5 \(\times\) 685 + 2 3427 = 7 \(\times\) 489 + 4 3427 = 11 \(\times\) 311 + 6 3427 = 13 \(\times\) 263 + 8 3427 = 17 \(\times\) 201 + 10 3427 = 19 \(\times\) 180 + 7 3427 = 23 \(\times\) 149 + 0
\par\addvspace{.55em}

            3427 est divisible par 23 donc\emph{  3427 n'est pas un nombre premier}.
            }

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

{\color{green!60!black}  617 = 2 \(\times\) 308 + 1 617 = 3 \(\times\) 205 + 2 617 = 5 \(\times\) 123 + 2 617 = 7 \(\times\) 88 + 1 617 = 11 \(\times\) 56 + 1 617 = 13 \(\times\) 47 + 6 617 = 17 \(\times\) 36 + 5 617 = 19 \(\times\) 32 + 9 617 = 23 \(\times\) 26 + 19 617 = 29 \(\times\) 21 + 8
\par\addvspace{.55em}

            617 n'a pas d'autres diviseurs que 1 et 617  donc \emph{ 617 est un nombre premier.} }

}
\item {\color{red}
        15978 est-il premier ?

{\color{green!60!black}

        15978 est pair donc \emph{15978 n'est pas un nombre premier}.}

}

\item {\color{red}
        1655 est-il premier ?

{\color{green!60!black}

        1655 se termine par un 5, c'est un multiple de 5 donc \emph{1655 n'est pas un nombre premier}.}

}

\end{enumerate}
\end{document}
