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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 251 et 377 par deux multiples consécutifs de 21.
    }

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

{\color{green!60!black}
        Quel est le plus grand multiple de 12 inférieur à 255 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 8 supérieur à 153 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 12180 et 9750 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}
\(\dfrac{12180}{9750}\)
\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  2349 et 8704.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 2349 et 8704.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{2349}{8704}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 1715;  397;  323; 16214 }

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

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

{\color{green!60!black}
        Encadre 251 et 377 par deux multiples consécutifs de 21.
        }

On effectue la division euclidienne de 251 par 21 :

 
\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) {5};
  \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) {21};
  \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};
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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[] 251 = 21 \(\times\) 11 + 20 et 20 \(<\) 21
 \item[] 251 = 231 + 20
 \item[] {\color{red}
donc 231 \(<\) 251 \(<\) 252 (231 + 21)}
\end{itemize}

De même:

On effectue la division euclidienne de 377 par 21 :

 
\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) {7};
  \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) {21};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.100) {1};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.880) {7};
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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[] 377 = 21 \(\times\) 17 + 20 et 20 \(<\) 21
 \item[] 377 = 357 + 20
 \item[] {\color{red}
donc 357 \(<\) 377 \(<\) 378 (357 + 21)}
\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 12 inférieur à 255 ?
           }

On effectue la division euclidienne de 255 par 12 :

 
\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) {5};
  \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) {12};
  \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) {2};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.100) {5};
  \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) {2};
  \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[] 255 = 12 \(\times\) 21 + 3 et 3 \(<\) 12
\item[] {\color{red} Le plus grand multiple de 12 inférieur à 255 est donc 12 \(\times\) 21 = 252.}
\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 8 supérieur à 153 ?
           }

On effectue la division euclidienne de 153 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) {1};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.540) {5};
  \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) {8};
  \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) {9};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {8};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.100) {3};
  \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) {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) {1};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 153 = 8 \(\times\) 19 + 1 et 1 \(<\) 8
\item[] {\color{red} Le plus petit multiple de 8 inférieur à 153 est donc 8 \(\times\) 20 = 160.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  12180 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
12180 & 2\\
6090 & 2\\
3045 & 3\\
1015 & 5\\
203 & 7\\
29 & 29\\
1 & \\
\end{tabular}\qquad 12180 = 2\textsuperscript{2} $\times$ 3 $\times$ 5 $\times$ 7 $\times$ 29
\end{center}}

    Décomposition de  9750 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
9750 & 2\\
4875 & 3\\
1625 & 5\\
325 & 5\\
65 & 5\\
13 & 13\\
1 & \\
\end{tabular}\qquad 9750 = 2 $\times$ 3 $\times$ 5\textsuperscript{3} $\times$ 13
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 12180 = 2\textsuperscript{2} \(\times\) 3 \(\times\) 5 \(\times\) 7 \(\times\) 29}
\par\addvspace{.55em}
{\color{green!60!black} 9750 = 2 \(\times\) 3 \(\times\) 5\textsuperscript{3} \(\times\) 13}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(12180;9750) = 2\textsuperscript{2} \(\times\) 3 \(\times\) 5\textsuperscript{3} \(\times\) 7 \(\times\) 13 \(\times\) 29 = 3958500}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(12180;9750) = 2 \(\times\) 3 \(\times\) 5 = 30}}
\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 \(\dfrac{12180}{9750}\) = \(\dfrac{12180:30}{9750:30}\) = \(\dfrac{406}{325}\) } }
\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} 2349 : \{ 1; 3; 9; 27; 29; 81; 87; 261; 783; 2349 \}
\par\addvspace{.55em}

8704 : \{ 1; 2; 4; 8; 16; 17; 32; 34; 64; 68; 128; 136; 256; 272; 512; 544; 1088; 2176; 4352; 8704 \}}

}
 \item {\color{red} Les diviseurs communs de 2349 et 8704 sont :

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

 }
\item {\color{red} Le plus grand diviseur commun de 2349 et 8704 est :

{\color{green!60!black}
PGCD(2349;8704) = 1}

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

{\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
2349 et 8704 sont premiers entre eux.}

 {\color{green!60!black} Donc la fraction \(\dfrac{2349}{8704}\) est irréductible.}}
\end{enumerate}
}

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

\begin{enumerate}
\item {\color{red}
        1715 est-il premier ?

{\color{green!60!black}

        1715 se termine par un 5, c'est un multiple de 5 donc \emph{1715 n'est pas un nombre premier}.}

}

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

{\color{green!60!black}  397 = 2 \(\times\) 198 + 1 397 = 3 \(\times\) 132 + 1 397 = 5 \(\times\) 79 + 2 397 = 7 \(\times\) 56 + 5 397 = 11 \(\times\) 36 + 1 397 = 13 \(\times\) 30 + 7 397 = 17 \(\times\) 23 + 6 397 = 19 \(\times\) 20 + 17 397 = 23 \(\times\) 17 + 6
\par\addvspace{.55em}

            397 n'a pas d'autres diviseurs que 1 et 397  donc \emph{ 397 est un nombre premier.} }

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

{\color{green!60!black}  323 = 2 \(\times\) 161 + 1 323 = 3 \(\times\) 107 + 2 323 = 5 \(\times\) 64 + 3 323 = 7 \(\times\) 46 + 1 323 = 11 \(\times\) 29 + 4 323 = 13 \(\times\) 24 + 11 323 = 17 \(\times\) 19 + 0
\par\addvspace{.55em}

            323 est divisible par 17 donc\emph{  323 n'est pas un nombre premier}.
            }

}
\item {\color{red}
        16214 est-il premier ?

{\color{green!60!black}

        16214 est pair donc \emph{16214 n'est pas un nombre premier}.}

}

\end{enumerate}
\end{document}
