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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 364 et 937 par deux multiples consécutifs de 22.
    }

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

{\color{green!60!black}
        Quel est le plus grand multiple de 15 inférieur à 220 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 12 supérieur à 201 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 5586 et 176 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}
\(\dfrac{5586}{176}\)
\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  520 et 729.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 520 et 729.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{520}{729}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 4141;  5913;  18152; 421 }

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

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

{\color{green!60!black}
        Encadre 364 et 937 par deux multiples consécutifs de 22.
        }

On effectue la division euclidienne de 364 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] (1.045,0.825) -- (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) {6};
  \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) {22};
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  \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[] 364 = 22 \(\times\) 16 + 12 et 12 \(<\) 22
 \item[] 364 = 352 + 12
 \item[] {\color{red}
donc 352 \(<\) 364 \(<\) 374 (352 + 22)}
\end{itemize}

De même:

On effectue la division euclidienne de 937 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);
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  \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[] 937 = 22 \(\times\) 42 + 13 et 13 \(<\) 22
 \item[] 937 = 924 + 13
 \item[] {\color{red}
donc 924 \(<\) 937 \(<\) 946 (924 + 22)}
\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 15 inférieur à 220 ?
           }

On effectue la division euclidienne de 220 par 15 :

 
\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) {2};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.540) {0};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {15};
  \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) {4};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {5};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.320) {1};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.100) {0};
  \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) {0};
  \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) {0};
  \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[] 220 = 15 \(\times\) 14 + 10 et 10 \(<\) 15
\item[] {\color{red} Le plus grand multiple de 15 inférieur à 220 est donc 15 \(\times\) 14 = 210.}
\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 12 supérieur à 201 ?
           }

On effectue la division euclidienne de 201 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) {0};
  \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) {12};
  \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) {2};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.320) {1};
  \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) {8};
  \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) {9};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 201 = 12 \(\times\) 16 + 9 et 9 \(<\) 12
\item[] {\color{red} Le plus petit multiple de 12 inférieur à 201 est donc 12 \(\times\) 17 = 204.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  5586 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
5586 & 2\\
2793 & 3\\
931 & 7\\
133 & 7\\
19 & 19\\
1 & \\
\end{tabular}\qquad 5586 = 2 $\times$ 3 $\times$ 7\textsuperscript{2} $\times$ 19
\end{center}}

    Décomposition de  176 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
176 & 2\\
88 & 2\\
44 & 2\\
22 & 2\\
11 & 11\\
1 & \\
\end{tabular}\qquad 176 = 2\textsuperscript{4} $\times$ 11
\end{center}}

\begin{enumerate}
\item {\color{red} Décompositions :
\par\addvspace{.55em}
{\color{green!60!black} 5586 = 2 \(\times\) 3 \(\times\) 7\textsuperscript{2} \(\times\) 19}
\par\addvspace{.55em}
{\color{green!60!black} 176 = 2\textsuperscript{4} \(\times\) 11}}
\item {\color{red} Calcul du plus petit multiple commun :
\par\addvspace{.55em}
{\color{green!60!black} PPCM(5586;176) = 2\textsuperscript{4} \(\times\) 3 \(\times\) 7\textsuperscript{2} \(\times\) 11 \(\times\) 19 = 491568}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(5586;176) = 2 = 2}}
\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 \(\dfrac{5586}{176}\) = \(\dfrac{5586:2}{176:2}\) = \(\dfrac{2793}{88}\) } }
\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} 520 : \{ 1; 2; 4; 5; 8; 10; 13; 20; 26; 40; 52; 65; 104; 130; 260; 520 \}
\par\addvspace{.55em}

729 : \{ 1; 3; 9; 27; 81; 243; 729 \}}

}
 \item {\color{red} Les diviseurs communs de 520 et 729 sont :

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

 }
\item {\color{red} Le plus grand diviseur commun de 520 et 729 est :

{\color{green!60!black}
PGCD(520;729) = 1}

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

{\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
520 et 729 sont premiers entre eux.}

 {\color{green!60!black} Donc la fraction \(\dfrac{520}{729}\) est irréductible.}}
\end{enumerate}
}

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

\begin{enumerate}
\item {\color{red}
        4141 est-il premier ?

{\color{green!60!black}  4141 = 2 \(\times\) 2070 + 1 4141 = 3 \(\times\) 1380 + 1 4141 = 5 \(\times\) 828 + 1 4141 = 7 \(\times\) 591 + 4 4141 = 11 \(\times\) 376 + 5 4141 = 13 \(\times\) 318 + 7 4141 = 17 \(\times\) 243 + 10 4141 = 19 \(\times\) 217 + 18 4141 = 23 \(\times\) 180 + 1 4141 = 29 \(\times\) 142 + 23 4141 = 31 \(\times\) 133 + 18 4141 = 37 \(\times\) 111 + 34 4141 = 41 \(\times\) 101 + 0
\par\addvspace{.55em}

            4141 est divisible par 41 donc\emph{  4141 n'est pas un nombre premier}.
            }

}
\item {\color{red}
        5913 est-il premier ?

{\color{green!60!black}

          5+9+1+3 = 18
\par\addvspace{.55em}
 1+8 = 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 5913 est divisible par 3.\emph{ donc 5913 n'est pas un nombre premier}.}

}
\item {\color{red}
        18152 est-il premier ?

{\color{green!60!black}

        18152 est pair donc \emph{18152 n'est pas un nombre premier}.}

}

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

{\color{green!60!black}  421 = 2 \(\times\) 210 + 1 421 = 3 \(\times\) 140 + 1 421 = 5 \(\times\) 84 + 1 421 = 7 \(\times\) 60 + 1 421 = 11 \(\times\) 38 + 3 421 = 13 \(\times\) 32 + 5 421 = 17 \(\times\) 24 + 13 421 = 19 \(\times\) 22 + 3 421 = 23 \(\times\) 18 + 7
\par\addvspace{.55em}

            421 n'a pas d'autres diviseurs que 1 et 421  donc \emph{ 421 est un nombre premier.} }

}
\end{enumerate}
\end{document}
