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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 659 et 724 par deux multiples consécutifs de 6.
    }

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

{\color{green!60!black}
        Quel est le plus grand multiple de 22 inférieur à 479 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 26 supérieur à 287 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 1664 et 2184 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}
\(\dfrac{1664}{2184}\)
\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  1216 et 15525.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 1216 et 15525.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{1216}{15525}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 677;  2641;  1845; 4779 }

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

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

{\color{green!60!black}
        Encadre 659 et 724 par deux multiples consécutifs de 6.
        }

On effectue la division euclidienne de 659 par 6 :

 
\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) {6};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.980) {5};
  \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) {6};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {5};
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\begin{itemize}
\item[] 659 = 6 \(\times\) 109 + 5 et 5 \(<\) 6
 \item[] 659 = 654 + 5
 \item[] {\color{red}
donc 654 \(<\) 659 \(<\) 660 (654 + 6)}
\end{itemize}

De même:

On effectue la division euclidienne de 724 par 6 :

 
\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) {7};
  \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) {4};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.980) {6};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.650) {1};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.420,1.650) {2};
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  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {2};
  \node[green!55!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) {4};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 724 = 6 \(\times\) 120 + 4 et 4 \(<\) 6
 \item[] 724 = 720 + 4
 \item[] {\color{red}
donc 720 \(<\) 724 \(<\) 726 (720 + 6)}
\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 22 inférieur à 479 ?
           }

On effectue la division euclidienne de 479 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] (1.265,0.825) -- (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) {4};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {7};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.540) {9};
  \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) {2};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.420,1.210) {1};
  \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) {4};
  \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) {3};
  \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) {2};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {7};
  \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[] 479 = 22 \(\times\) 21 + 17 et 17 \(<\) 22
\item[] {\color{red} Le plus grand multiple de 22 inférieur à 479 est donc 22 \(\times\) 21 = 462.}
\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 26 supérieur à 287 ?
           }

On effectue la division euclidienne de 287 par 26 :

 
\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) {2};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {8};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.540) {7};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {26};
  \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) {1};
  \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) {2};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.100) {7};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {2};
  \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) {2};
  \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[] 287 = 26 \(\times\) 11 + 1 et 1 \(<\) 26
\item[] {\color{red} Le plus petit multiple de 26 inférieur à 287 est donc 26 \(\times\) 12 = 312.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  1664 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
1664 & 2\\
832 & 2\\
416 & 2\\
208 & 2\\
104 & 2\\
52 & 2\\
26 & 2\\
13 & 13\\
1 & \\
\end{tabular}\qquad 1664 = 2\textsuperscript{7} $\times$ 13
\end{center}}

    Décomposition de  2184 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
2184 & 2\\
1092 & 2\\
546 & 2\\
273 & 3\\
91 & 7\\
13 & 13\\
1 & \\
\end{tabular}\qquad 2184 = 2\textsuperscript{3} $\times$ 3 $\times$ 7 $\times$ 13
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 1664 = 2\textsuperscript{7} \(\times\) 13}
\par\addvspace{.55em}
{\color{green!60!black} 2184 = 2\textsuperscript{3} \(\times\) 3 \(\times\) 7 \(\times\) 13}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(1664;2184) = 2\textsuperscript{7} \(\times\) 3 \(\times\) 7 \(\times\) 13 = 34944}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(1664;2184) = 2\textsuperscript{3} \(\times\) 13 = 104}}
\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 \(\dfrac{1664}{2184}\) = \(\dfrac{1664:104}{2184:104}\) = \(\dfrac{16}{21}\) } }
\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} 1216 : \{ 1; 2; 4; 8; 16; 19; 32; 38; 64; 76; 152; 304; 608; 1216 \}
\par\addvspace{.55em}

15525 : \{ 1; 3; 5; 9; 15; 23; 25; 27; 45; 69; 75; 115; 135; 207; 225; 345; 575; 621; 675; 1035; 1725; 3105; 5175; 15525 \}}

}
 \item {\color{red} Les diviseurs communs de 1216 et 15525 sont :

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

 }
\item {\color{red} Le plus grand diviseur commun de 1216 et 15525 est :

{\color{green!60!black}
PGCD(1216;15525) = 1}

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

{\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
1216 et 15525 sont premiers entre eux.}

 {\color{green!60!black} Donc la fraction \(\dfrac{1216}{15525}\) est irréductible.}}
\end{enumerate}
}

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

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

{\color{green!60!black}  677 = 2 \(\times\) 338 + 1 677 = 3 \(\times\) 225 + 2 677 = 5 \(\times\) 135 + 2 677 = 7 \(\times\) 96 + 5 677 = 11 \(\times\) 61 + 6 677 = 13 \(\times\) 52 + 1 677 = 17 \(\times\) 39 + 14 677 = 19 \(\times\) 35 + 12 677 = 23 \(\times\) 29 + 10 677 = 29 \(\times\) 23 + 10
\par\addvspace{.55em}

            677 n'a pas d'autres diviseurs que 1 et 677  donc \emph{ 677 est un nombre premier.} }

}
\item {\color{red}
        2641 est-il premier ?

{\color{green!60!black}  2641 = 2 \(\times\) 1320 + 1 2641 = 3 \(\times\) 880 + 1 2641 = 5 \(\times\) 528 + 1 2641 = 7 \(\times\) 377 + 2 2641 = 11 \(\times\) 240 + 1 2641 = 13 \(\times\) 203 + 2 2641 = 17 \(\times\) 155 + 6 2641 = 19 \(\times\) 139 + 0
\par\addvspace{.55em}

            2641 est divisible par 19 donc\emph{  2641 n'est pas un nombre premier}.
            }

}
\item {\color{red}
        1845 est-il premier ?

{\color{green!60!black}

        1845 se termine par un 5, c'est un multiple de 5 donc \emph{1845 n'est pas un nombre premier}.}

}

\item {\color{red}
        4779 est-il premier ?

{\color{green!60!black}

          4+7+7+9 = 27
\par\addvspace{.55em}
 2+7 = 9

{\color{green!60!black} Critère de divisibilité par 3 :
\par\addvspace{.55em}

                     Un nombre entier est divisible par 3 si (et seulement si) la somme de ses chiffres est elle-même divisible par 3.
                     .}

                     donc 4779 est divisible par 3.\emph{ donc 4779 n'est pas un nombre premier}.}

}
\end{enumerate}
\end{document}
