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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 506 et 819 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 24 inférieur à 568 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 6 supérieur à 86 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 2205 et 6600 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}
\(\dfrac{2205}{6600}\)
\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  176 et 9765.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 176 et 9765.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{176}{9765}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 14694;  631;  2197; 855 }

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

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

{\color{green!60!black}
        Encadre 506 et 819 par deux multiples consécutifs de 8.
        }

On effectue la division euclidienne de 506 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) {5};
  \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) {6};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {8};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.210) {6};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.420,1.210) {3};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {2};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 506 = 8 \(\times\) 63 + 2 et 2 \(<\) 8
 \item[] 506 = 504 + 2
 \item[] {\color{red}
donc 504 \(<\) 506 \(<\) 512 (504 + 8)}
\end{itemize}

De même:

On effectue la division euclidienne de 819 par 8 :

 
\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) {1};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.980) {9};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.980) {8};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.540) {0};
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  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.100) {9};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {1};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.880) {6};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.880) {1};
  \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[] 819 = 8 \(\times\) 102 + 3 et 3 \(<\) 8
 \item[] 819 = 816 + 3
 \item[] {\color{red}
donc 816 \(<\) 819 \(<\) 824 (816 + 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 24 inférieur à 568 ?
           }

On effectue la division euclidienne de 568 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) {5};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {6};
  \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) {24};
  \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) {3};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {8};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.320) {4};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.100) {8};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {8};
  \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) {7};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {6};
  \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[] 568 = 24 \(\times\) 23 + 16 et 16 \(<\) 24
\item[] {\color{red} Le plus grand multiple de 24 inférieur à 568 est donc 24 \(\times\) 23 = 552.}
\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 6 supérieur à 86 ?
           }

On effectue la division euclidienne de 86 par 6 :

 
\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) {8};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {6};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.980,1.540) {6};
  \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) {4};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.320) {6};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {6};
  \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) {4};
  \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[] 86 = 6 \(\times\) 14 + 2 et 2 \(<\) 6
\item[] {\color{red} Le plus petit multiple de 6 inférieur à 86 est donc 6 \(\times\) 15 = 90.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  2205 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
2205 & 3\\
735 & 3\\
245 & 5\\
49 & 7\\
7 & 7\\
1 & \\
\end{tabular}\qquad 2205 = 3\textsuperscript{2} $\times$ 5 $\times$ 7\textsuperscript{2}
\end{center}}

    Décomposition de  6600 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
6600 & 2\\
3300 & 2\\
1650 & 2\\
825 & 3\\
275 & 5\\
55 & 5\\
11 & 11\\
1 & \\
\end{tabular}\qquad 6600 = 2\textsuperscript{3} $\times$ 3 $\times$ 5\textsuperscript{2} $\times$ 11
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 2205 = 3\textsuperscript{2} \(\times\) 5 \(\times\) 7\textsuperscript{2}}
\par\addvspace{.55em}
{\color{green!60!black} 6600 = 2\textsuperscript{3} \(\times\) 3 \(\times\) 5\textsuperscript{2} \(\times\) 11}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(2205;6600) = 2\textsuperscript{3} \(\times\) 3\textsuperscript{2} \(\times\) 5\textsuperscript{2} \(\times\) 7\textsuperscript{2} \(\times\) 11 = 970200}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(2205;6600) = 3 \(\times\) 5 = 15}}
\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 \(\dfrac{2205}{6600}\) = \(\dfrac{2205:15}{6600:15}\) = \(\dfrac{147}{440}\) } }
\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} 176 : \{ 1; 2; 4; 8; 11; 16; 22; 44; 88; 176 \}
\par\addvspace{.55em}

9765 : \{ 1; 3; 5; 7; 9; 15; 21; 31; 35; 45; 63; 93; 105; 155; 217; 279; 315; 465; 651; 1085; 1395; 1953; 3255; 9765 \}}

}
 \item {\color{red} Les diviseurs communs de 176 et 9765 sont :

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

 }
\item {\color{red} Le plus grand diviseur commun de 176 et 9765 est :

{\color{green!60!black}
PGCD(176;9765) = 1}

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

{\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
176 et 9765 sont premiers entre eux.}

 {\color{green!60!black} Donc la fraction \(\dfrac{176}{9765}\) est irréductible.}}
\end{enumerate}
}

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

\begin{enumerate}
\item {\color{red}
        14694 est-il premier ?

{\color{green!60!black}

        14694 est pair donc \emph{14694 n'est pas un nombre premier}.}

}

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

{\color{green!60!black}  631 = 2 \(\times\) 315 + 1 631 = 3 \(\times\) 210 + 1 631 = 5 \(\times\) 126 + 1 631 = 7 \(\times\) 90 + 1 631 = 11 \(\times\) 57 + 4 631 = 13 \(\times\) 48 + 7 631 = 17 \(\times\) 37 + 2 631 = 19 \(\times\) 33 + 4 631 = 23 \(\times\) 27 + 10 631 = 29 \(\times\) 21 + 22
\par\addvspace{.55em}

            631 n'a pas d'autres diviseurs que 1 et 631  donc \emph{ 631 est un nombre premier.} }

}
\item {\color{red}
        2197 est-il premier ?

{\color{green!60!black}  2197 = 2 \(\times\) 1098 + 1 2197 = 3 \(\times\) 732 + 1 2197 = 5 \(\times\) 439 + 2 2197 = 7 \(\times\) 313 + 6 2197 = 11 \(\times\) 199 + 8 2197 = 13 \(\times\) 169 + 0
\par\addvspace{.55em}

            2197 est divisible par 13 donc\emph{  2197 n'est pas un nombre premier}.
            }

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

{\color{green!60!black}

        855 se termine par un 5, c'est un multiple de 5 donc \emph{855 n'est pas un nombre premier}.}

}

\end{enumerate}
\end{document}
