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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 93 et 386 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 26 inférieur à 513 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 21 supérieur à 494 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 15960 et 4140 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}
\(\dfrac{15960}{4140}\)
\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  15200 et 8019.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 15200 et 8019.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{15200}{8019}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 607;  4301;  975; 15784 }

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

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

{\color{green!60!black}
        Encadre 93 et 386 par deux multiples consécutifs de 8.
        }

On effectue la division euclidienne de 93 par 8 :

 
\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) {9};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {3};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.980,1.540) {8};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.980,1.210) {1};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.605) {5};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 93 = 8 \(\times\) 11 + 5 et 5 \(<\) 8
 \item[] 93 = 88 + 5
 \item[] {\color{red}
donc 88 \(<\) 93 \(<\) 96 (88 + 8)}
\end{itemize}

De même:

On effectue la division euclidienne de 386 par 8 :

 
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  \draw[gray] (1.540,1.430) -- (1.540,1.320);
  \draw[black] (1.980,1.705) -- (1.980,0.220);
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  \draw[black] (1.265,0.825) -- (1.595,0.825);
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  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {8};
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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[] 386 = 8 \(\times\) 48 + 2 et 2 \(<\) 8
 \item[] 386 = 384 + 2
 \item[] {\color{red}
donc 384 \(<\) 386 \(<\) 392 (384 + 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 26 inférieur à 513 ?
           }

On effectue la division euclidienne de 513 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);
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  \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 (1.540,1.540) {3};
  \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};
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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 (1.540,0.605) {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[] 513 = 26 \(\times\) 19 + 19 et 19 \(<\) 26
\item[] {\color{red} Le plus grand multiple de 26 inférieur à 513 est donc 26 \(\times\) 19 = 494.}
\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 21 supérieur à 494 ?
           }

On effectue la division euclidienne de 494 par 21 :

 
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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] (1.265,0.825) -- (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) {4};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {9};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.540) {4};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {21};
  \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) {2};
  \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) {4};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {7};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.880) {3};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.880) {6};
  \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[] 494 = 21 \(\times\) 23 + 11 et 11 \(<\) 21
\item[] {\color{red} Le plus petit multiple de 21 inférieur à 494 est donc 21 \(\times\) 24 = 504.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  15960 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
15960 & 2\\
7980 & 2\\
3990 & 2\\
1995 & 3\\
665 & 5\\
133 & 7\\
19 & 19\\
1 & \\
\end{tabular}\qquad 15960 = 2\textsuperscript{3} $\times$ 3 $\times$ 5 $\times$ 7 $\times$ 19
\end{center}}

    Décomposition de  4140 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
4140 & 2\\
2070 & 2\\
1035 & 3\\
345 & 3\\
115 & 5\\
23 & 23\\
1 & \\
\end{tabular}\qquad 4140 = 2\textsuperscript{2} $\times$ 3\textsuperscript{2} $\times$ 5 $\times$ 23
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 15960 = 2\textsuperscript{3} \(\times\) 3 \(\times\) 5 \(\times\) 7 \(\times\) 19}
\par\addvspace{.55em}
{\color{green!60!black} 4140 = 2\textsuperscript{2} \(\times\) 3\textsuperscript{2} \(\times\) 5 \(\times\) 23}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(15960;4140) = 2\textsuperscript{3} \(\times\) 3\textsuperscript{2} \(\times\) 5 \(\times\) 7 \(\times\) 19 \(\times\) 23 = 1101240}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(15960;4140) = 2\textsuperscript{2} \(\times\) 3 \(\times\) 5 = 60}}
\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 \(\dfrac{15960}{4140}\) = \(\dfrac{15960:60}{4140:60}\) = \(\dfrac{266}{69}\) } }
\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} 15200 : \{ 1; 2; 4; 5; 8; 10; 16; 19; 20; 25; 32; 38; 40; 50; 76; 80; 95; 100; 152; 160; 190; 200; 304; 380; 400; 475; 608; 760; 800; 950; 1520; 1900; 3040; 3800; 7600; 15200 \}
\par\addvspace{.55em}

8019 : \{ 1; 3; 9; 11; 27; 33; 81; 99; 243; 297; 729; 891; 2673; 8019 \}}

}
 \item {\color{red} Les diviseurs communs de 15200 et 8019 sont :

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

 }
\item {\color{red} Le plus grand diviseur commun de 15200 et 8019 est :

{\color{green!60!black}
PGCD(15200;8019) = 1}

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

{\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
15200 et 8019 sont premiers entre eux.}

 {\color{green!60!black} Donc la fraction \(\dfrac{15200}{8019}\) est irréductible.}}
\end{enumerate}
}

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

\begin{enumerate}
\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.} }

}
\item {\color{red}
        4301 est-il premier ?

{\color{green!60!black}  4301 = 2 \(\times\) 2150 + 1 4301 = 3 \(\times\) 1433 + 2 4301 = 5 \(\times\) 860 + 1 4301 = 7 \(\times\) 614 + 3 4301 = 11 \(\times\) 391 + 0
\par\addvspace{.55em}

            4301 est divisible par 11 donc\emph{  4301 n'est pas un nombre premier}.
            }

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

{\color{green!60!black}

        975 se termine par un 5, c'est un multiple de 5 donc \emph{975 n'est pas un nombre premier}.}

}

\item {\color{red}
        15784 est-il premier ?

{\color{green!60!black}

        15784 est pair donc \emph{15784 n'est pas un nombre premier}.}

}

\end{enumerate}
\end{document}
