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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 935 et 385 par deux multiples consécutifs de 24.
    }

\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 407 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 9 supérieur à 121 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 2448 et 13312 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}
\(\dfrac{2448}{13312}\)
\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  3125 et 408.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 3125 et 408.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{3125}{408}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 4503;  1667;  10008; 3959 }

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

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

{\color{green!60!black}
        Encadre 935 et 385 par deux multiples consécutifs de 24.
        }

On effectue la division euclidienne de 935 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] (1.045,0.825) -- (1.595,0.825);
  \draw[blue] (0.990,1.705) .. controls (1.137,1.778) and (1.283,1.778) .. (1.430,1.705);
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.540) {9};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {3};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.540) {5};
  \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) {3};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.420,1.210) {8};
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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[] 935 = 24 \(\times\) 38 + 23 et 23 \(<\) 24
 \item[] 935 = 912 + 23
 \item[] {\color{red}
donc 912 \(<\) 935 \(<\) 936 (912 + 24)}
\end{itemize}

De même:

On effectue la division euclidienne de 385 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] (1.045,0.825) -- (1.595,0.825);
  \draw[blue] (0.990,1.705) .. controls (1.137,1.778) and (1.283,1.778) .. (1.430,1.705);
  \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) {8};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.540) {5};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {24};
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  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.420,1.210) {6};
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  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.320) {2};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {4};
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  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.880) {4};
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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[] 385 = 24 \(\times\) 16 + 1 et 1 \(<\) 24
 \item[] 385 = 384 + 1
 \item[] {\color{red}
donc 384 \(<\) 385 \(<\) 408 (384 + 24)}
\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 407 ?
           }

On effectue la division euclidienne de 407 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);
  \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) {0};
  \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) {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) {2};
  \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};
  \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) {4};
  \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) {3};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {1};
  \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[] 407 = 18 \(\times\) 22 + 11 et 11 \(<\) 18
\item[] {\color{red} Le plus grand multiple de 18 inférieur à 407 est donc 18 \(\times\) 22 = 396.}
\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 9 supérieur à 121 ?
           }

On effectue la division euclidienne de 121 par 9 :

 
\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) {1};
  \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) {9};
  \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) {3};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {9};
  \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) {3};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.880) {7};
  \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) {4};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 121 = 9 \(\times\) 13 + 4 et 4 \(<\) 9
\item[] {\color{red} Le plus petit multiple de 9 inférieur à 121 est donc 9 \(\times\) 14 = 126.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  2448 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
2448 & 2\\
1224 & 2\\
612 & 2\\
306 & 2\\
153 & 3\\
51 & 3\\
17 & 17\\
1 & \\
\end{tabular}\qquad 2448 = 2\textsuperscript{4} $\times$ 3\textsuperscript{2} $\times$ 17
\end{center}}

    Décomposition de  13312 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
13312 & 2\\
6656 & 2\\
3328 & 2\\
1664 & 2\\
832 & 2\\
416 & 2\\
208 & 2\\
104 & 2\\
52 & 2\\
26 & 2\\
13 & 13\\
1 & \\
\end{tabular}\qquad 13312 = 2\textsuperscript{10} $\times$ 13
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 2448 = 2\textsuperscript{4} \(\times\) 3\textsuperscript{2} \(\times\) 17}
\par\addvspace{.55em}
{\color{green!60!black} 13312 = 2\textsuperscript{10} \(\times\) 13}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(2448;13312) = 2\textsuperscript{10} \(\times\) 3\textsuperscript{2} \(\times\) 13 \(\times\) 17 = 2036736}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(2448;13312) = 2\textsuperscript{4} = 16}}
\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 \(\dfrac{2448}{13312}\) = \(\dfrac{2448:16}{13312:16}\) = \(\dfrac{153}{832}\) } }
\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} 3125 : \{ 1; 5; 25; 125; 625; 3125 \}
\par\addvspace{.55em}

408 : \{ 1; 2; 3; 4; 6; 8; 12; 17; 24; 34; 51; 68; 102; 136; 204; 408 \}}

}
 \item {\color{red} Les diviseurs communs de 3125 et 408 sont :

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

 }
\item {\color{red} Le plus grand diviseur commun de 3125 et 408 est :

{\color{green!60!black}
PGCD(3125;408) = 1}

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

{\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
3125 et 408 sont premiers entre eux.}

 {\color{green!60!black} Donc la fraction \(\dfrac{3125}{408}\) est irréductible.}}
\end{enumerate}
}

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

\begin{enumerate}
\item {\color{red}
        4503 est-il premier ?

{\color{green!60!black}

          4+5+0+3 = 12
\par\addvspace{.55em}
 1+2 = 3

{\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 4503 est divisible par 3.\emph{ donc 4503 n'est pas un nombre premier}.}

}
\item {\color{red}
        1667 est-il premier ?

{\color{green!60!black}  1667 = 2 \(\times\) 833 + 1 1667 = 3 \(\times\) 555 + 2 1667 = 5 \(\times\) 333 + 2 1667 = 7 \(\times\) 238 + 1 1667 = 11 \(\times\) 151 + 6 1667 = 13 \(\times\) 128 + 3 1667 = 17 \(\times\) 98 + 1 1667 = 19 \(\times\) 87 + 14 1667 = 23 \(\times\) 72 + 11 1667 = 29 \(\times\) 57 + 14 1667 = 31 \(\times\) 53 + 24 1667 = 37 \(\times\) 45 + 2 1667 = 41 \(\times\) 40 + 27
\par\addvspace{.55em}

            1667 n'a pas d'autres diviseurs que 1 et 1667  donc \emph{ 1667 est un nombre premier.} }

}
\item {\color{red}
        10008 est-il premier ?

{\color{green!60!black}

        10008 est pair donc \emph{10008 n'est pas un nombre premier}.}

}

\item {\color{red}
        3959 est-il premier ?

{\color{green!60!black}  3959 = 2 \(\times\) 1979 + 1 3959 = 3 \(\times\) 1319 + 2 3959 = 5 \(\times\) 791 + 4 3959 = 7 \(\times\) 565 + 4 3959 = 11 \(\times\) 359 + 10 3959 = 13 \(\times\) 304 + 7 3959 = 17 \(\times\) 232 + 15 3959 = 19 \(\times\) 208 + 7 3959 = 23 \(\times\) 172 + 3 3959 = 29 \(\times\) 136 + 15 3959 = 31 \(\times\) 127 + 22 3959 = 37 \(\times\) 107 + 0
\par\addvspace{.55em}

            3959 est divisible par 37 donc\emph{  3959 n'est pas un nombre premier}.
            }

}
\end{enumerate}
\end{document}
