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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 965 et 316 par deux multiples consécutifs de 7.
    }

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

{\color{green!60!black}
        Quel est le plus grand multiple de 27 inférieur à 256 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 23 supérieur à 389 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 108 et 15925 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}
\(\dfrac{108}{15925}\)
\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  88 et 627.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 88 et 627.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{88}{627}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 1015;  113;  16068; 3799 }

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

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

{\color{green!60!black}
        Encadre 965 et 316 par deux multiples consécutifs de 7.
        }

On effectue la division euclidienne de 965 par 7 :

 
\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) {9};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.980) {6};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.980) {5};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.980) {7};
  \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) {3};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.640,1.650) {7};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.760) {7};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {6};
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\begin{itemize}
\item[] 965 = 7 \(\times\) 137 + 6 et 6 \(<\) 7
 \item[] 965 = 959 + 6
 \item[] {\color{red}
donc 959 \(<\) 965 \(<\) 966 (959 + 7)}
\end{itemize}

De même:

On effectue la division euclidienne de 316 par 7 :

 
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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);
  \draw[blue] (0.990,1.705) .. controls (1.137,1.778) and (1.283,1.778) .. (1.430,1.705);
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  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {7};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.210) {4};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.420,1.210) {5};
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  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.605) {1};
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\begin{itemize}
\item[] 316 = 7 \(\times\) 45 + 1 et 1 \(<\) 7
 \item[] 316 = 315 + 1
 \item[] {\color{red}
donc 315 \(<\) 316 \(<\) 322 (315 + 7)}
\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 27 inférieur à 256 ?
           }

On effectue la division euclidienne de 256 par 27 :

 
\begin{center}\begin{tikzpicture}[line width=0.6pt]
  \draw[black] (1.980,1.265) -- (1.980,0.220);
  \draw[black] (1.980,0.990) -- (2.860,0.990);
  \draw[black] (1.045,0.825) -- (1.595,0.825);
  \draw[blue] (0.990,1.265) .. controls (1.210,1.338) and (1.430,1.338) .. (1.650,1.265);
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.100) {2};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {5};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.100) {6};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.100) {27};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,0.770) {9};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,0.880) {3};
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  \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 (1.540,0.605) {3};
  \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[] 256 = 27 \(\times\) 9 + 13 et 13 \(<\) 27
\item[] {\color{red} Le plus grand multiple de 27 inférieur à 256 est donc 27 \(\times\) 9 = 243.}
\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 23 supérieur à 389 ?
           }

On effectue la division euclidienne de 389 par 23 :

 
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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] (1.045,0.825) -- (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) {9};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {23};
  \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) {6};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {3};
  \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) {9};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {5};
  \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) {8};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,0.880) {3};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,0.880) {1};
  \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) {2};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 389 = 23 \(\times\) 16 + 21 et 21 \(<\) 23
\item[] {\color{red} Le plus petit multiple de 23 inférieur à 389 est donc 23 \(\times\) 17 = 391.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  108 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
108 & 2\\
54 & 2\\
27 & 3\\
9 & 3\\
3 & 3\\
1 & \\
\end{tabular}\qquad 108 = 2\textsuperscript{2} $\times$ 3\textsuperscript{3}
\end{center}}

    Décomposition de  15925 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
15925 & 5\\
3185 & 5\\
637 & 7\\
91 & 7\\
13 & 13\\
1 & \\
\end{tabular}\qquad 15925 = 5\textsuperscript{2} $\times$ 7\textsuperscript{2} $\times$ 13
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 108 = 2\textsuperscript{2} \(\times\) 3\textsuperscript{3}}
\par\addvspace{.55em}
{\color{green!60!black} 15925 = 5\textsuperscript{2} \(\times\) 7\textsuperscript{2} \(\times\) 13}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(108;15925) = 2\textsuperscript{2} \(\times\) 3\textsuperscript{3} \(\times\) 5\textsuperscript{2} \(\times\) 7\textsuperscript{2} \(\times\) 13 = 1719900}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(108,15925) =  1}}
 \item {\color{red} Simplification de la fraction :
\par\addvspace{.55em}
 {\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
108 et 15925 sont premiers entre eux.
\par\addvspace{.55em}

         Donc la fraction \(\dfrac{108}{15925}\) est irréductible.}}
\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} 88 : \{ 1; 2; 4; 8; 11; 22; 44; 88 \}
\par\addvspace{.55em}

627 : \{ 1; 3; 11; 19; 33; 57; 209; 627 \}}

}
 \item {\color{red} Les diviseurs communs de 88 et 627 sont :

{\color{green!60!black}
\{ 1; 11 \}}

 }
\item {\color{red} Le plus grand diviseur commun de 88 et 627 est :

{\color{green!60!black}
PGCD(88;627) = 11}

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

{\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 \(\dfrac{88}{627}\) = \(\dfrac{88:11}{627:11}\) = \(\dfrac{8}{57}\) } }
\end{enumerate}
}

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

\begin{enumerate}
\item {\color{red}
        1015 est-il premier ?

{\color{green!60!black}

        1015 se termine par un 5, c'est un multiple de 5 donc \emph{1015 n'est pas un nombre premier}.}

}

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

{\color{green!60!black}  113 = 2 \(\times\) 56 + 1 113 = 3 \(\times\) 37 + 2 113 = 5 \(\times\) 22 + 3 113 = 7 \(\times\) 16 + 1 113 = 11 \(\times\) 10 + 3
\par\addvspace{.55em}

            113 n'a pas d'autres diviseurs que 1 et 113  donc \emph{ 113 est un nombre premier.} }

}
\item {\color{red}
        16068 est-il premier ?

{\color{green!60!black}

        16068 est pair donc \emph{16068 n'est pas un nombre premier}.}

}

\item {\color{red}
        3799 est-il premier ?

{\color{green!60!black}  3799 = 2 \(\times\) 1899 + 1 3799 = 3 \(\times\) 1266 + 1 3799 = 5 \(\times\) 759 + 4 3799 = 7 \(\times\) 542 + 5 3799 = 11 \(\times\) 345 + 4 3799 = 13 \(\times\) 292 + 3 3799 = 17 \(\times\) 223 + 8 3799 = 19 \(\times\) 199 + 18 3799 = 23 \(\times\) 165 + 4 3799 = 29 \(\times\) 131 + 0
\par\addvspace{.55em}

            3799 est divisible par 29 donc\emph{  3799 n'est pas un nombre premier}.
            }

}
\end{enumerate}
\end{document}
