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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 822 et 288 par deux multiples consécutifs de 7.
    }

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

{\color{green!60!black}
        Quel est le plus grand multiple de 24 inférieur à 301 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 5 supérieur à 77 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 9261 et 2816 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}
\(\dfrac{9261}{2816}\)
\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  14210 et 5589.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 14210 et 5589.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{14210}{5589}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 475;  19484;  1943; 607 }

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

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

{\color{green!60!black}
        Encadre 822 et 288 par deux multiples consécutifs de 7.
        }

On effectue la division euclidienne de 822 par 7 :

 
\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[gray] (1.320,1.870) -- (1.320,1.760);
  \draw[gray] (1.540,1.870) -- (1.540,1.320);
  \draw[black] (1.980,2.145) -- (1.980,0.220);
  \draw[black] (1.980,1.870) -- (3.080,1.870);
  \draw[black] (1.045,1.705) -- (1.155,1.705);
  \draw[black] (1.045,1.265) -- (1.375,1.265);
  \draw[black] (1.265,0.825) -- (1.595,0.825);
  \draw[blue] (0.990,2.145) .. controls (1.063,2.218) and (1.137,2.218) .. (1.210,2.145);
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.980) {8};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.980) {2};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.980) {2};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.980) {7};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.650) {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[] 822 = 7 \(\times\) 117 + 3 et 3 \(<\) 7
 \item[] 822 = 819 + 3
 \item[] {\color{red}
donc 819 \(<\) 822 \(<\) 826 (819 + 7)}
\end{itemize}

De même:

On effectue la division euclidienne de 288 par 7 :

 
\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);
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {1};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 288 = 7 \(\times\) 41 + 1 et 1 \(<\) 7
 \item[] 288 = 287 + 1
 \item[] {\color{red}
donc 287 \(<\) 288 \(<\) 294 (287 + 7)}
\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 24 inférieur à 301 ?
           }

On effectue la division euclidienne de 301 par 24 :

 
\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) {0};
  \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) {24};
  \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) {2};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {4};
  \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) {6};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.880) {8};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.880) {4};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {3};
  \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[] 301 = 24 \(\times\) 12 + 13 et 13 \(<\) 24
\item[] {\color{red} Le plus grand multiple de 24 inférieur à 301 est donc 24 \(\times\) 12 = 288.}
\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 5 supérieur à 77 ?
           }

On effectue la division euclidienne de 77 par 5 :

 
\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[gray] (1.320,1.430) -- (1.320,1.320);
  \draw[black] (1.760,1.705) -- (1.760,0.220);
  \draw[black] (1.760,1.430) -- (2.640,1.430);
  \draw[black] (1.045,1.265) -- (1.155,1.265);
  \draw[black] (1.045,0.825) -- (1.375,0.825);
  \draw[blue] (0.990,1.705) .. controls (1.063,1.778) and (1.137,1.778) .. (1.210,1.705);
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.540) {7};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {7};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.980,1.540) {5};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.980,1.210) {1};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.210) {5};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.320) {5};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {7};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.100) {2};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.880) {5};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,0.880) {2};
  \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[] 77 = 5 \(\times\) 15 + 2 et 2 \(<\) 5
\item[] {\color{red} Le plus petit multiple de 5 inférieur à 77 est donc 5 \(\times\) 16 = 80.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  9261 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
9261 & 3\\
3087 & 3\\
1029 & 3\\
343 & 7\\
49 & 7\\
7 & 7\\
1 & \\
\end{tabular}\qquad 9261 = 3\textsuperscript{3} $\times$ 7\textsuperscript{3}
\end{center}}

    Décomposition de  2816 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
2816 & 2\\
1408 & 2\\
704 & 2\\
352 & 2\\
176 & 2\\
88 & 2\\
44 & 2\\
22 & 2\\
11 & 11\\
1 & \\
\end{tabular}\qquad 2816 = 2\textsuperscript{8} $\times$ 11
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 9261 = 3\textsuperscript{3} \(\times\) 7\textsuperscript{3}}
\par\addvspace{.55em}
{\color{green!60!black} 2816 = 2\textsuperscript{8} \(\times\) 11}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(9261;2816) = 2\textsuperscript{8} \(\times\) 3\textsuperscript{3} \(\times\) 7\textsuperscript{3} \(\times\) 11 = 26078976}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(9261,2816) =  1}}
 \item {\color{red} Simplification de la fraction :
\par\addvspace{.55em}
 {\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
9261 et 2816 sont premiers entre eux.
\par\addvspace{.55em}

         Donc la fraction \(\dfrac{9261}{2816}\) 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} 14210 : \{ 1; 2; 5; 7; 10; 14; 29; 35; 49; 58; 70; 98; 145; 203; 245; 290; 406; 490; 1015; 1421; 2030; 2842; 7105; 14210 \}
\par\addvspace{.55em}

5589 : \{ 1; 3; 9; 23; 27; 69; 81; 207; 243; 621; 1863; 5589 \}}

}
 \item {\color{red} Les diviseurs communs de 14210 et 5589 sont :

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

 }
\item {\color{red} Le plus grand diviseur commun de 14210 et 5589 est :

{\color{green!60!black}
PGCD(14210;5589) = 1}

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

{\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
14210 et 5589 sont premiers entre eux.}

 {\color{green!60!black} Donc la fraction \(\dfrac{14210}{5589}\) est irréductible.}}
\end{enumerate}
}

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

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

{\color{green!60!black}

        475 se termine par un 5, c'est un multiple de 5 donc \emph{475 n'est pas un nombre premier}.}

}

\item {\color{red}
        19484 est-il premier ?

{\color{green!60!black}

        19484 est pair donc \emph{19484 n'est pas un nombre premier}.}

}

\item {\color{red}
        1943 est-il premier ?

{\color{green!60!black}  1943 = 2 \(\times\) 971 + 1 1943 = 3 \(\times\) 647 + 2 1943 = 5 \(\times\) 388 + 3 1943 = 7 \(\times\) 277 + 4 1943 = 11 \(\times\) 176 + 7 1943 = 13 \(\times\) 149 + 6 1943 = 17 \(\times\) 114 + 5 1943 = 19 \(\times\) 102 + 5 1943 = 23 \(\times\) 84 + 11 1943 = 29 \(\times\) 67 + 0
\par\addvspace{.55em}

            1943 est divisible par 29 donc\emph{  1943 n'est pas un nombre premier}.
            }

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

{\color{green!60!black}  607 = 2 \(\times\) 303 + 1 607 = 3 \(\times\) 202 + 1 607 = 5 \(\times\) 121 + 2 607 = 7 \(\times\) 86 + 5 607 = 11 \(\times\) 55 + 2 607 = 13 \(\times\) 46 + 9 607 = 17 \(\times\) 35 + 12 607 = 19 \(\times\) 31 + 18 607 = 23 \(\times\) 26 + 9 607 = 29 \(\times\) 20 + 27
\par\addvspace{.55em}

            607 n'a pas d'autres diviseurs que 1 et 607  donc \emph{ 607 est un nombre premier.} }

}
\end{enumerate}
\end{document}
