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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 184 et 536 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 20 inférieur à 411 ?
        }

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

{\color{green!60!black}
        Quel est le plus petit multiple de 26 supérieur à 358 ?
        }

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

{\color{green!60!black}
\begin{enumerate}
\item Décompose 4560 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}
\(\dfrac{4560}{4140}\)
\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  5481 et 3125.
 \item Donne la liste ordonnée de leurs diviseurs communs.
 \item Déduis-en le PGCD de 5481 et 3125.
 \item Simplifie au maximum la fraction :
\par\addvspace{.55em}
\(\dfrac{5481}{3125}\)
\end{enumerate}
}

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

{\color{green!60!black} Les nombres suivants sont-ils premiers ? 1265;  10124;  479; 2533 }

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

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

{\color{green!60!black}
        Encadre 184 et 536 par deux multiples consécutifs de 21.
        }

On effectue la division euclidienne de 184 par 21 :

 
\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) {1};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {8};
  \node[black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.540,1.100) {4};
  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.100) {21};
  \node[blue, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,0.770) {8};
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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[] 184 = 21 \(\times\) 8 + 16 et 16 \(<\) 21
 \item[] 184 = 168 + 16
 \item[] {\color{red}
donc 168 \(<\) 184 \(<\) 189 (168 + 21)}
\end{itemize}

De même:

On effectue la division euclidienne de 536 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] (1.045,0.825) -- (1.595,0.825);
  \draw[blue] (0.990,1.705) .. controls (1.137,1.778) and (1.283,1.778) .. (1.430,1.705);
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  \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) {6};
  \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) {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};
  \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[] 536 = 21 \(\times\) 25 + 11 et 11 \(<\) 21
 \item[] 536 = 525 + 11
 \item[] {\color{red}
donc 525 \(<\) 536 \(<\) 546 (525 + 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 20 inférieur à 411 ?
           }

On effectue la division euclidienne de 411 par 20 :

 
\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) {1};
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  \node[black, anchor=base west, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (2.200,1.540) {20};
  \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) {0};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.320) {0};
  \node[red, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.100,1.320) {4};
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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[] 411 = 20 \(\times\) 20 + 11 et 11 \(<\) 20
\item[] {\color{red} Le plus grand multiple de 20 inférieur à 411 est donc 20 \(\times\) 20 = 400.}
\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 26 supérieur à 358 ?
           }

On effectue la division euclidienne de 358 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);
  \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) {3};
  \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) {8};
  \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};
  \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) {6};
  \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) {8};
  \node[green!55!black, anchor=base, font=\ttfamily \fontsize{6.3}{7.5}\selectfont] at (1.320,1.100) {9};
  \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) {7};
  \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) {2};
\end{tikzpicture}\end{center}


\begin{itemize}
\item[] 358 = 26 \(\times\) 13 + 20 et 20 \(<\) 26
\item[] {\color{red} Le plus petit multiple de 26 inférieur à 358 est donc 26 \(\times\) 14 = 364.}
\end{itemize}

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

{\color{green!60!black}
    Décomposition de  4560 en produit de facteurs premiers :

    
{\color{black}\begin{center}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r|l}
4560 & 2\\
2280 & 2\\
1140 & 2\\
570 & 2\\
285 & 3\\
95 & 5\\
19 & 19\\
1 & \\
\end{tabular}\qquad 4560 = 2\textsuperscript{4} $\times$ 3 $\times$ 5 $\times$ 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} 4560 = 2\textsuperscript{4} \(\times\) 3 \(\times\) 5 \(\times\) 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(4560;4140) = 2\textsuperscript{4} \(\times\) 3\textsuperscript{2} \(\times\) 5 \(\times\) 19 \(\times\) 23 = 314640}}
\item {\color{red} Calcul du plus grand
         diviseur commun :
\par\addvspace{.55em}
{\color{green!60!black} PGCD(4560;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 \(\dfrac{4560}{4140}\) = \(\dfrac{4560:60}{4140:60}\) = \(\dfrac{76}{69}\) } }
\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} 5481 : \{ 1; 3; 7; 9; 21; 27; 29; 63; 87; 189; 203; 261; 609; 783; 1827; 5481 \}
\par\addvspace{.55em}

3125 : \{ 1; 5; 25; 125; 625; 3125 \}}

}
 \item {\color{red} Les diviseurs communs de 5481 et 3125 sont :

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

 }
\item {\color{red} Le plus grand diviseur commun de 5481 et 3125 est :

{\color{green!60!black}
PGCD(5481;3125) = 1}

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

{\color{green!60!black} Leurs PGCD étant égal à 1, les nombres
5481 et 3125 sont premiers entre eux.}

 {\color{green!60!black} Donc la fraction \(\dfrac{5481}{3125}\) est irréductible.}}
\end{enumerate}
}

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

\begin{enumerate}
\item {\color{red}
        1265 est-il premier ?

{\color{green!60!black}

        1265 se termine par un 5, c'est un multiple de 5 donc \emph{1265 n'est pas un nombre premier}.}

}

\item {\color{red}
        10124 est-il premier ?

{\color{green!60!black}

        10124 est pair donc \emph{10124 n'est pas un nombre premier}.}

}

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

{\color{green!60!black}  479 = 2 \(\times\) 239 + 1 479 = 3 \(\times\) 159 + 2 479 = 5 \(\times\) 95 + 4 479 = 7 \(\times\) 68 + 3 479 = 11 \(\times\) 43 + 6 479 = 13 \(\times\) 36 + 11 479 = 17 \(\times\) 28 + 3 479 = 19 \(\times\) 25 + 4 479 = 23 \(\times\) 20 + 19
\par\addvspace{.55em}

            479 n'a pas d'autres diviseurs que 1 et 479  donc \emph{ 479 est un nombre premier.} }

}
\item {\color{red}
        2533 est-il premier ?

{\color{green!60!black}  2533 = 2 \(\times\) 1266 + 1 2533 = 3 \(\times\) 844 + 1 2533 = 5 \(\times\) 506 + 3 2533 = 7 \(\times\) 361 + 6 2533 = 11 \(\times\) 230 + 3 2533 = 13 \(\times\) 194 + 11 2533 = 17 \(\times\) 149 + 0
\par\addvspace{.55em}

            2533 est divisible par 17 donc\emph{  2533 n'est pas un nombre premier}.
            }

}
\end{enumerate}
\end{document}
