% proflycee-tools-arithm.tex
% Copyright 2023-2024  Cédric Pierquet
% Released under the LaTeX Project Public License v1.3c or later, see http://www.latex-project.org/lppl.txt

\RequirePackage{ifthen}
\RequirePackage{modulus}

%%------ConversionsBases
%dec->bin avec blocs de 4 chiffres
\setKVdefault[CONVDECBIN]{%
	AffBase=true
}
\NewDocumentCommand\ConversionDecBin{ s O{} m }{%
	\useKVdefault[CONVDECBIN]
	\setKV[CONVDECBIN]{#2}% on paramètres les nouvelles clés et on les simplifie
	\def\resbrut{\xintDecToBin{#3}}%
	\StrLen{\resbrut}[\nbchiffres]%
	\def\nbgrp{\fpeval{4*ceil(\nbchiffres/4,0)}}%
	\IfBooleanTF{#1}%
		{\num{#3}\ifboolKV[CONVDECBIN]{AffBase}{_{10}}{}=\num[digit-group-size=4]{\resbrut}\ifboolKV[CONVDECBIN]{AffBase}{_{2}}{}}%
		{\num{#3}\ifboolKV[CONVDECBIN]{AffBase}{_{10}}{}=\num[digit-group-size=4,minimum-integer-digits=\nbgrp]{\resbrut}\ifboolKV[CONVDECBIN]{AffBase}{_{2}}{}}%
}

\setKVdefault[CONVBINHEX]{%
	%Epaisseur=0.75pt,%
	AffBase=true,%
	Details=true
}
%bourrage de 0 avant
\ExplSyntaxOn
\NewExpandableDocumentCommand{\PLstrzeros}{m}
{
	\int_compare:nT { #1 > 0 }
	{
		0 \prg_replicate:nn { #1 - 1 } { 0 }
	}
}
\ExplSyntaxOff


%la conversion complète
\newcommand\ConversionBinHex[2][]{%
	\useKVdefault[CONVBINHEX]%
	\setKV[CONVBINHEX]{#1}% on paramètres les nouvelles clés et on les simplifie
	\def\chbrut{#2}%
	\StrLen{\chbrut}[\nbchiffres]%nb de chiffres du binaire
	\xdef\nbgrp{\fpeval{4*ceil(\nbchiffres/4,0)}} %nb de chiffres avec blocs de 4
	\xdef\nbblocs{\fpeval{\nbgrp/4}} %nb de blocs
	%on rajoute des zeros si besoin := OK
	\xdef\resinter{\chbrut}%
	\num[digit-group-size=4]{\chbrut}\ifboolKV[CONVBINHEX]{AffBase}{_{2}}{}=%
	\ifboolKV[CONVBINHEX]{Details}{%
	\ifnum\nbchiffres<\nbgrp%
		\xdef\nbz{\inteval{\nbgrp-\nbchiffres}}%
		\xdef\resinter{\PLstrzeros{\nbz}\chbrut}%
		\num[digit-group-size=4,minimum-integer-digits=\nbgrp]{\resinter}=%
	\fi%
	%découpage par blocs et conversion en hexa := OK
	\newcount\cpt%
	\cpt0%
	\loop\ifnum \cpt<\nbblocs%
		\def\iinit{\fpeval{4*\cpt+1}}%
		\def\ifinal{\fpeval{4*(\cpt+1)}}%
		\StrMid{\resinter}{\iinit}{\ifinal}[\blocinter]%
		{\underbracket{\blocinter}_{\xintBinToHex{\blocinter}}\,}%
		\advance\cpt by 1%
	\repeat%
	\!=%
	}%
	{}%
	\xintBinToHex{\chbrut}\ifboolKV[CONVBINHEX]{AffBase}{_{16}}{}%
}

%hexa-bin par bloc de 4
\setKVdefault[CONVHEXBIN]{%
	%Epaisseur=0.75pt,
	AffBase=true,%
	Details=true
}

\newcommand\ConvHexBinBloc[1]{%
	\def\binbrut{\xintHexToBin{#1}}%
	\StrLen{\binbrut}[\nbchiffresbinbrut]%nb
	\PLstrzeros{\xinteval{4-\nbchiffresbinbrut}}{\binbrut}%
}

\newcommand\ConversionHexBin[2][]{%
	\useKVdefault[CONVHEXBIN]%
	\setKV[CONVHEXBIN]{#1}% on paramètres les nouvelles clés et on les simplifie
	%\def\thicktraitshexbin{\useKV[CONVHEXBIN]{Epaisseur}}%
	\ifboolKV[CONVHEXBIN]{AffBase}{#2_{16}=}{#2=}%
	\StrLen{#2}[\nbchiffreshex]%nb
	\ifboolKV[CONVHEXBIN]{Details}%
	{%
		\foreach \i in {1,...,\nbchiffreshex}{%
			\StrChar{#2}{\i}[\tmpcharhex]%
			{\underbracket{\ConvHexBinBloc{\tmpcharhex}}_{\tmpcharhex}\,}
		}%
		\ifboolKV[CONVHEXBIN]{AffBase}{{}_{2}}{}%
	}%
	{%
		\foreach \i in {1,...,\nbchiffreshex}{%
			\StrChar{#2}{\i}[\tmpcharhex]%
			\ConvHexBinBloc{\tmpcharhex}\,%
		}%
		\!\ifboolKV[CONVHEXBIN]{AffBase}{{}_{2}}{}%
	}%
}


%hexa/bin->dec avec écriture polynomiale
\defKV[CONVTODEC]{%
	BaseDep=\def\basedepart{#1}
}
\setKVdefault[CONVTODEC]{%
	BaseDep=2,%
	AffBase=true,%
	Details=true,%
	Zeros=true
}

\ExplSyntaxOn
\newcommand\convertbasetobasedix[2]{%
	\int_from_base:nn {#1}{#2}
}
\ExplSyntaxOff

\newcommand\ConversionVersDec[2][]{%
	\useKVdefault[CONVTODEC]
	\setKV[CONVTODEC]{#1}% on paramètres les nouvelles clés et on les simplifie
	\def\nbdepart{#2}%
	\StrLen{\nbdepart}[\nbchiffres]%
	\StrChar{\nbdepart}{1}[\chiffre]%
	%si on est en base 16
	\xintifboolexpr{\basedepart == 16}%
	{%
		\nbdepart\ifboolKV[CONVTODEC]{AffBase}{_{\basedepart}}{} =%
		\ifboolKV[CONVTODEC]{Details}{%
			\xintHexToDec{\chiffre}\times\basedepart^{\inteval{\nbchiffres-1}}%
			\newcount\cpt%
			\cpt2%
			\loop\ifnum \cpt<\inteval{\nbchiffres+1}%
				\def\puiss{\inteval{\nbchiffres-\cpt}}%
				\StrChar{\nbdepart}{\cpt}[\chiffre]%
				\ifboolKV[CONVTODEC]{Zeros}%
					{%
						+\xintHexToDec{\chiffre}\times\basedepart^{\puiss}%
					}%
					{%
						\ifnum\xintHexToDec{\chiffre} > 0%
							+\xintHexToDec{\chiffre}\times\basedepart^{\puiss}%
						\fi%
					}%
				\advance\cpt by 1%
			\repeat%
			=%
			}%
			{}%
		\num{\xintHexToDec{\nbdepart}}\ifboolKV[CONVTODEC]{AffBase}{_{10}}{}%
	}%
	{}%
	\xintifboolexpr{\basedepart == 2}%
	{%
		\num[digit-group-size=4]{\nbdepart}\ifboolKV[CONVTODEC]{AffBase}{_{\basedepart}}{} =%
		\ifboolKV[CONVTODEC]{Details}{%
			\chiffre\times\basedepart^{\inteval{\nbchiffres-1}}%
			\newcount\cpt%
			\cpt2%
			\loop\ifnum \cpt<\inteval{\nbchiffres+1}%
				\def\puiss{\inteval{\nbchiffres-\cpt}}%
				\StrChar{\nbdepart}{\cpt}[\chiffre]%
				\ifboolKV[CONVTODEC]{Zeros}%
				{%
					+\chiffre\times\basedepart^{\puiss}%
				}
				{
					\ifnum\chiffre > 0%
						+\chiffre\times\basedepart^{\puiss}%
					\fi%
				}%
			\advance\cpt by 1%
			\repeat%
			=%
			}%
			{}%
		\num{\xintBinToDec{\nbdepart}}\ifboolKV[CONVTODEC]{AffBase}{_{10}}{}%
	}%
	{}%
}

\newcommand\ConversionBaseDix[3][]{%1=options,%2=nb,%3=basedep ??
	\useKVdefault[CONVTODEC]
	\setKV[CONVTODEC]{#1}% on paramètres les nouvelles clés et on les simplifie
	\def\NBdepart{#2}%
	\def\basedepart{#3}%
	\StrLen{\NBdepart}[\nbchiffres]%
	\StrChar{\NBdepart}{1}[\chiffre]%
	\NBdepart\ifboolKV[CONVTODEC]{AffBase}{_{\basedepart}}{} =%
	\ifboolKV[CONVTODEC]{Details}{%
		\xintHexToDec{\chiffre}\times\basedepart^{\inteval{\nbchiffres-1}}%
		\newcount\cpt%
		\cpt2%
		\loop\ifnum \cpt<\inteval{\nbchiffres+1}%
			\def\puiss{\inteval{\nbchiffres-\cpt}}%
			\StrChar{\NBdepart}{\cpt}[\chiffre]%
			\ifboolKV[CONVTODEC]{Zeros}%
				{%
					+\xintHexToDec{\chiffre}\times\basedepart^{\puiss}%
				}%
				{%
					\ifnum\xintHexToDec{\chiffre} > 0%
						+\xintHexToDec{\chiffre}\times\basedepart^{\puiss}%
					\fi%
				}%
			\advance\cpt by 1%
		\repeat%
		=%
		}%
		{}%
	\num{\convertbasetobasedix{#2}{#3}}\ifboolKV[CONVTODEC]{AffBase}{_{10}}{}%
}

%%------CONVFROMDEC
\newcommand\PLnoeud[2]{\tikz[remember picture,baseline=(#1.base)]\node[shape=rectangle,inner sep=0pt](#1){#2};}
\ExplSyntaxOn
\newcommand\convertbasedixtobase[2]{%
	\int_to_Base:nn {#1}{#2}
}
\ExplSyntaxOff

\defKV[convfromten]{%
	Couleur=\def\PLConvCouleur{#1},%
	DecalH=\def\PLConvDecalH{#1},%
	DecalV=\def\PLConvDecalV{#1},%
	Noeud=\def\PLConvNoeud{#1}
}

\setKVdefault[convfromten]{%
	Couleur=red,%
	DecalH=2pt,%
	DecalV=3pt,%
	Rect=true,%
	Noeud=EEE,%
	CouleurRes=false
}

\newcommand\ConversionDepuisBaseDix[3][]{%
	\useKVdefault[convfromten]%
	\setKV[convfromten]{#1}%
	\xdef\ValRes{\xintDecToHex{#2}}%
	\xdef\ValA{#2}\xdef\ValB{#3}%
	\xdef\ValTMP{#2}%
	\xdef\ValMU{\inteval{#3-1}}%
	\ensuremath{%
		\left\lbrace\begin{array}{@{\,}r@{\;=\;}l@{\;+\;}r}
			%1ere division
			\xdef\ValQ{\fpeval{trunc(\ValTMP/#3,0)}}\xdef\ValR{\fpeval{\ValTMP-#3*\ValQ}}
			\num{\ValTMP}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}\PLnoeud{\PLConvNoeud1}{\num{\ValR}}%
			\xdef\ValTMP{\ValQ}%
			\whiledo {\ValTMP > \ValMU}%
			{%
				\xdef\ValQ{\fpeval{trunc(\ValTMP/#3,0)}}\xdef\ValR{\fpeval{\ValTMP-#3*\ValQ}}%
				\\ \num{\ValTMP}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}\num{\ValR}
				\xdef\ValTMP{\ValQ}%
			}
			%dernière
			\xdef\ValQ{\fpeval{trunc(\ValTMP/#3,0)}}\xdef\ValR{\fpeval{\ValTMP-#3*\ValQ}}%
			\\ \num{\ValTMP}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}\PLnoeud{\PLConvNoeud2}{\num{\ValR}}%
		\end{array} \right| \Rightarrow \num{#2}_{10}=\ifboolKV[convfromten]{CouleurRes}{\mathcolor{\PLConvCouleur}{\convertbasedixtobase{#2}{#3}_{#3}}}{\convertbasedixtobase{#2}{#3}_{#3}}}%
	\ifboolKV[convfromten]{Rect}%
	{%
		\IfSubStr{\PLConvDecalH}{/}%
			{\StrCut{\PLConvDecalH}{/}{\PLConvDecalHg}{\PLConvDecalHd}}%
			{\def\PLConvDecalHg{\PLConvDecalH}\def\PLConvDecalHd{\PLConvDecalH}}%
		\begin{tikzpicture}
			\draw[overlay,rounded corners=4pt,\PLConvCouleur,thick] ($(\PLConvNoeud1.north west)+(-\PLConvDecalHg,\PLConvDecalV)$) rectangle ($(\PLConvNoeud2.south east)+(\PLConvDecalHd,-\PLConvDecalV)$) ;
			\draw[overlay,rounded corners=4pt,\PLConvCouleur,thick,->,>=latex] ($(\PLConvNoeud2.east)+(\PLConvDecalHd,0)$)--++(0,{0.75\baselineskip}) ;
		\end{tikzpicture}%
	}{}%
}

%%------PRESPGCD
\DeclareMathOperator{\PLpgcd}{PGCD}
\defKV[prespgcd]{%
	Couleur=\def\PLPGCDCouleur{#1},%
	DecalRect=\def\PLPGCDDecal{#1},%
	Noeud=\def\PLPGCDNoeud{#1}
}

\setKVdefault[prespgcd]{%
	Couleur=red,%
	DecalRect=2pt,%
	Rectangle=true,%
	Noeud=FFF,%
	CouleurResultat=false,%
	AfficheConclusion=true,%
	AfficheDelimiteurs=true
}

\RequirePackage{xintgcd}

\newcommand\PresentationPGCD[3][]{%
	\useKVdefault[prespgcd]%
	\setKV[prespgcd]{#1}%
	\xdef\respgcd{\xinteval{gcd(#2,#3)}}
	\xdef\ValA{#2}\xdef\ValB{#3}%on stocke les valeurs du départ
	\ensuremath{%
		\ifboolKV[prespgcd]{AfficheDelimiteurs}%
		{\left\lbrace}%
		{}%
		\begin{array}{@{\,}r@{\;=\;}l@{\;+\;}r}
			%1ère division
			\xdef\ValQ{\fpeval{trunc(\ValA/\ValB,0)}}\xdef\ValR{\fpeval{\ValA-\ValB*\ValQ}}%
			\num{\ValA}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}%
			\xintifboolexpr{\ValR == \respgcd}%
			{\PLnoeud{\PLPGCDNoeud1}{\num{\ValR}}}%noeud si c'est le pgcd
			{\num{\ValR}}%
			\xdef\ValA{\ValB}\xdef\ValB{\ValR}%nouvelles valeurs
			\whiledo {\ValR > 0}%
			{%
				\xdef\ValQ{\fpeval{trunc(\ValA/\ValB,0)}}\xdef\ValR{\fpeval{\ValA-\ValB*\ValQ}}%
				\\%
				\num{\ValA}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}%
				\xintifboolexpr{\ValR == \respgcd}%
				{\PLnoeud{\PLPGCDNoeud1}{\num{\ValR}}}%noeud si c'est le pgcd
				{\num{\ValR}}%
				\xdef\ValA{\ValB}\xdef\ValB{\ValR}%nouvelles valeurs
			}%
		\end{array}%
		\ifboolKV[prespgcd]{AfficheDelimiteurs}%
		{\right|}%
		{}%
		\ifboolKV[prespgcd]{AfficheConclusion}%
		{%
			\Rightarrow \PLpgcd\left(\num{#2};\num{#3}\right)=\ifboolKV[prespgcd]{CouleurResultat}{\mathcolor{\PLPGCDCouleur}{\num{\respgcd}}}{\num{\respgcd}}%
		}%
		{}%
	}%
	\ifboolKV[prespgcd]{Rectangle}%
	{%
		\begin{tikzpicture}
			\draw[overlay,rounded corners=4pt,\PLPGCDCouleur,thick] ($(\PLPGCDNoeud1.north west)+(-\PLPGCDDecal,\PLPGCDDecal)$) rectangle ($(\PLPGCDNoeud1.south east)+(\PLPGCDDecal,-\PLPGCDDecal)$) ;
		\end{tikzpicture}%
	}{}%
}

%%===égalité de Bezout
\NewDocumentCommand\AffCoeffBezout{ m }{%
	\xintifboolexpr{#1 < 0}%
		{\left( \num{#1} \right)}%
		{\num{#1}}%
}
\NewDocumentCommand\EgaliteBezout{ O{black} m m }{%
	\xintAssign{\xintBezout{#2}{#3}}\to\TmpU\TmpV\TmpD%
	\ensuremath{\num{#2} \times \mathcolor{#1}{\AffCoeffBezout{\TmpU}} + \AffCoeffBezout{#3} \times \mathcolor{#1}{\AffCoeffBezout{\TmpV}} = \num{\TmpD}}%
}

%%===Équations diophantiennes
\RequirePackage[thicklines]{cancel}%comme PfC
\NewDocumentCommand\AffCoeffDioph{ m }{%
	\xintifboolexpr{#1 < 0}%
		{\left( \num{#1} \right)}%
		{\num{#1}}%
}
\NewDocumentCommand\AffCoeffDiophSign{ m }{%
	\xintifboolexpr{#1 < 0}%
		{\num{#1}}%
		{+\num{#1}}%
}

\defKV[eqdioph]{%
	Lettre=\def\LettreSolEDioph{#1},%
	Couleur=\def\CouleurSolEDioph{#1},%
	Inconnues=\def\InconnuesSolEDioph{#1},%
	Entier=\def\KKK{#1}
}

\setKVdefault[eqdioph]{%
	Lettre=E,%
	Couleur=black,%
	Inconnues=x/y,%
	Entier=k,%
	Cadres=false,%
	PresPGCD=true
}

\NewDocumentCommand\EquationDiophantienne{ O{} m }{%v2 avec équation en "dur"
	\useKVdefault[eqdioph]%
	\setKV[eqdioph]{#1}%
	\setlength{\parindent}{0pt}%
	%extractions des paramètres
	\StrBefore[1]{\InconnuesSolEDioph}{/}[\XXX]%
	\StrBehind[1]{\InconnuesSolEDioph}{/}[\YYY]%
	\StrBefore{#2}{\XXX}[\AA]%
	\StrBetween{#2}{\XXX}{\YYY}[\BB]%
	\StrBehind{#2}{=}[\CC]%
	\IfStrEq{\AA}{}%
		{\def\AA{1}}{}%
	\IfStrEq{\AA}{-}%
		{\def\AA{-1}}{}%
	\StrLen{\BB}[\lgtB]%
	\xintifboolexpr{ \lgtB > 1 }%+b ou -b
		{%
			\StrDel{\BB}{+}[\BB]%
		}%
		{%
			\IfStrEq{\BB}{-}%
				{\def\BB{-1}}{}%
			\IfStrEq{\BB}{+}%
				{\def\BB{1}}{}%
		}%
	%Calcul du PGCD
	\xdef\PGCDD{\xinteval{gcd(\AA,\BB)}}%
	On cherche à résoudre l'équation diophantienne :\[ \num{\AA}\XXX + \AffCoeffDioph{\BB}\YYY=\num{\CC} \xintifboolexpr{ \PGCDD == 1 'or' \xintiiRem{\CC}{\PGCDD} != 0 }{\qquad (\LettreSolEDioph)}{} \]%
	\ifboolKV[eqdioph]{PresPGCD}%
		{D'après l'algorithme d'Euclide : \PresentationPGCD[Rectangle=false]{\xinteval{abs(\AA)}}{\xinteval{abs(\BB)}}.}%
		{Le PGCD de \num{\AA} et de \num{\BB} vaut \num{\PGCDD}.}%
	\par\smallskip
	\xintifboolexpr{ \xintiiRem{\CC}{\PGCDD} == 0 }%solutions obligatoires
		{%
			\xintifboolexpr{ \PGCDD == 1}%
				{%
					Les entiers \num{\xinteval{abs(\AA)}} et \num{\xinteval{abs(\BB)}} sont premiers entre eux, donc l'équation $(\LettreSolEDioph)$ admet une infinité de solutions.\par
					\xdef\AAA{\AA}\xdef\BBB{\BB}\xdef\CCC{\CC}%
				}%
				{%
					Le PGCD de \num{\AA} et \num{\BB} divise \num{\CC}, donc on peut simplifier l'équation diophantienne par \num{\PGCDD}.%
					\xdef\AAA{\xintiieval{\AA/\PGCDD}}\xdef\BBB{\xintiieval{\BB/\PGCDD}}\xdef\CCC{\xintiieval{\CC/\PGCDD}}%
					%
					\[ \num{\AA}\XXX+\AffCoeffDioph{\BB}\YYY=\num{\CC} \underset{\div\num{\PGCDD}}{\Longleftrightarrow} \num{\AAA}\XXX+\AffCoeffDioph{\BBB}\YYY=\num{\CCC} \qquad (\LettreSolEDioph) \]%
					Les entiers \num{\AAA} et \num{\BBB} sont premiers entre eux, donc l'équation $(\LettreSolEDioph)$ admet une infinité de solutions.\par
				}%
			\xintAssign{\xintBezout{\AAA}{\BBB}}\to\TmpU\TmpV\TmpD
			%
			On détermine une solution particulière de $(E)$ : \[ \num{\AAA} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffBezout{\TmpU}} + \AffCoeffBezout{\BBB} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffBezout{\TmpV}} = \num{\TmpD}
			\xintifboolexpr{ \CCC != 1}%
				{%
					\underset{\times\AffCoeffDioph{\CCC}}{\implies}
					\num{\AAA} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\TmpU*\CCC}}} + \AffCoeffBezout{\BBB} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\TmpV*\CCC}}} = \num{\CCC}
				}%
				{}%
				\qquad ({\LettreSolEDioph}_0)
			 \]%
			%
			Par soustraction :
			%
			\[%
			{\renewcommand\arraystretch{1.25}%
				\begin{array}{ @{\,} c @{\,} c @{\;\times\;} c @{\;+\;} c @{\;\times\;} c @{\;=\;} c }
					& \num{\AAA} & \XXX & \AffCoeffDioph{\BBB} & \YYY & \num{\CCC} \\
					-~~~~~ & \num{\AAA} & \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\TmpU*\CCC}}} & \AffCoeffDioph{\BBB} & \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\TmpV*\CCC}}} & \num{\CCC} \\ \hline
					& \num{\AAA} & \left( \XXX \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\TmpU*\CCC}}} \right)& \AffCoeffDioph{\BBB} & \left( \YYY \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\TmpV*\CCC}}} \right) & 0\\
				\end{array}}
			\]%
			\def\TmpPartieA{\XXX \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\TmpU*\CCC}}}}%
			\def\TmpPartieB{\YYY \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\TmpV*\CCC}}}}%
			%
			On en déduit que $\num{\AAA} \times \underbrace{\left( \TmpPartieA \right)}_{\text{entier}} = \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$, et donc que $\num{\AAA} \mid \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$.\par\smallskip
			Or \num{\xinteval{abs(\AAA)}} et \num{\xinteval{abs(\BBB)}} sont premiers entre eux, donc d'après le théorème de Gauss, on a $\num{\AAA} \mid \TmpPartieB$.\par
			Il existe donc un entier $\KKK$ tel que $\TmpPartieB = \num{\AAA} \times \KKK$, ce qui donne 
			$\ifboolKV[eqdioph]{Cadres}
				{\boxed{\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}}
				{\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}
			$.\par
			En remplaçant, on obtient :
			%
			\begin{align*}
				\num{\AAA} \times \left( \TmpPartieA \right) = \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right) & \implies \num{\AAA} \times \left( \TmpPartieA \right) = \num{\xinteval{-\BBB}} \times \big( \underbrace{\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}} \AffCoeffDiophSign{\AAA}\KKK}_{\mathclap{\YYY}}  \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\CCC*\TmpV}}} \big) \\
				& \implies \num{\AAA} \times \left( \TmpPartieA \right) = \num{\xinteval{-\BBB}} \times \left( \num{\AAA}\KKK \right) \\
				& \implies \TmpPartieA = \num{\xinteval{-\BBB}}\KKK \\
				& \implies \ifboolKV[eqdioph]{Cadres}
					{\boxed{\XXX = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK}}
					{\XXX = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK}
			\end{align*}
			%
			Ainsi, si $\XXX$ et $\YYY$ sont solutions de $(\LettreSolEDioph)$, alors il existe un entier $\KKK$ tel que ${\XXX=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK}$ et ${\YYY=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}$.\par\medskip
			Réciproquement, soit $\KKK$ un entier quelconque :
			%
			\begin{align*}
				\num{\AAA} \times \left( \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \right) + \AffCoeffDioph{\BBB} \times  \left( \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK \right) & = \num{\AAA} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\CCC*\TmpU}}} + \cancel{\AffCoeffDioph{\AAA} \times \AffCoeffDioph{\xinteval{-\BBB}} \KKK} + \AffCoeffDioph{\BBB} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\CCC*\TmpV}}} +  \cancel{\AffCoeffDioph{\BBB} \times \AffCoeffDioph{\AAA} \KKK} \\
				& = \underbrace{\num{\AAA} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\CCC*\TmpU}}} +  \AffCoeffDioph{\BBB} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\CCC*\TmpV}}}}_{=\,\num{\CCC} \text{ d'après } ({\LettreSolEDioph}_0)} \\
				& = \num{\CCC}
			\end{align*}
			%
			On en déduit que $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK \right)$ est solution de $(\LettreSolEDioph)$.\par\medskip
			En conclusion, les solutions de $(E)$ sont donc les couples $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}} \AffCoeffDiophSign{\AAA}\KKK \right)$, avec $\KKK$ un entier relatif.
		}%
		{%
			Le PGCD de \num{\AA} et \num{\BB} ne divise pas \num{\CC}, donc l'équation $(\LettreSolEDioph)$ n'admet aucune solution.
		}%
}

%liste diviseurs
\setKVdefault[listdiv]{%
	AffNom=true
}
\NewDocumentCommand\ListeDiviseurs{ s O{} m }{%
	\useKVdefault[listdiv]%
	\setKV[listdiv]{#2}%
	\xdef\tmplistdiv{1}%
	\xdef\argcal{\xinteval{#3}}%
	\xintFor* ##1 in {\xintSeq{2}{\argcal}}\do{%
		\xintifboolexpr{ \xintiiRem{\argcal}{##1} == 0 }%
		{%
			\xdef\tmplistdiv{\tmplistdiv /\num{##1}}%
		}%
		{}%
	}%
	\ensuremath{\ifboolKV[listdiv]{AffNom}{\IfBooleanTF{#1}{\mathscr{D}}{\mathcal{D}}_{\num{\argcal}}=}{}\EcritureEnsemble[\strut]{\tmplistdiv}}%
}

%arbre diviseurs
\defKV[arbrediviseurs]{%
	EspaceNiveau=\def\TmpEspNiv{#1},%
	EspaceFeuille=\def\TMpEspFeuille{#1},%
	CouleurDetails=\def\TmpCoulDetails{#1},%
	Echelle=\def\TmpEchelle{#1}
}

\setKVdefault[arbrediviseurs]{%
	EspaceNiveau=2.25,%
	EspaceFeuille=0.66,%
	Details=true,%
	CouleurDetails=red,%
	Echelle=1,%
	Fleches=true
}

\NewDocumentCommand\ArbreDiviseurs{ O{} m }{%
	\useKVdefault[arbrediviseurs]%
	\setKV[arbrediviseurs]{#1}%
	% test avec CPoulain ^^
	\xdef\tmpcalc{\xinteval{#2}}%
	\xdef\tmparg{}%
	\newcount\anp\newcount\bnp\newcount\cnp%
	\newcount\pileb\newcount\exposant%
	\exposant=0\relax%
	\anp=\tmpcalc\relax%
	\bnp=2\relax%
	\pileb=2\relax%
	\whiledo{\the\anp > 1}{%
		\modulo{\the\anp}{\the\bnp}%
		\ifnum\remainder=0\relax%
		\cnp=\numexpr\anp/\bnp\relax%
		\exposant=\numexpr\exposant+1\relax%
		\anp=\cnp\relax%
		\else%
		\ifnum\exposant>0\relax%
		\xdef\tmparg{\tmparg\the\pileb,\the\exposant*}%
		%\expandafter\UpdatetoksCPier\Foo\nil%
		\fi%
		\bnp=\numexpr\bnp+1\relax%
		\pileb=\bnp\relax%
		\exposant=0\relax%
		\fi%
	}%
	\xdef\tmparg{\tmparg\the\bnp,{\the\exposant}}%
	%\expandafter\UpdatetoksCPier\the\bnp,{\the\exposant}\nil%
	% \xdef\argnum{\xinteval{#2}}%
	% \def\decompotmp{\DecompoPremierExposant{#2}}%
	\setsepchar{*/,}%
	\xdef\Foo{\tmparg}%
	\readlist*\readcaractdiv{\Foo}%
	% fin test CPoulain ^^
	\xdef\arbredivnbdivprem{\readcaractdivlen}%nombre diviseurs permiers
	\xdef\arbredivnbdiv{1}%
	\xintFor* ##1 in {\xintSeq{1}{\arbredivnbdivprem}}\do{\xdef\arbredivnbdiv{\xinteval{\arbredivnbdiv*(\readcaractdiv[##1,2]+1)}}}%nombre diviseurs
	\def\HauteurTotale{\xinteval{\arbredivnbdiv-1}}%
	\ifboolKV[arbrediviseurs]{Fleches}%
	{\tikzstyle{flechearbrediv} = [line width=\fpeval{\TmpEchelle*0.6}pt,->,>=latex]}%
	{\tikzstyle{flechearbrediv} = [line width=\fpeval{\TmpEchelle*0.6}pt]}%
	\begin{tikzpicture}[scale=\TmpEchelle,every node/.style={scale=\TmpEchelle}]
		% grille d'aide
		% \draw[xstep=\TmpEspNiv,ystep=\TMpEspFeuille,thin,lightgray] (0,{(-\arbredivnbdiv+1)*\TMpEspFeuille}) grid ({\arbredivnbdivprem*\TmpEspNiv},0) ;
		% dernier niveau OK
		\xdef\nbsommets{\arbredivnbdiv}%
		\xintFor* ##1 in {\xintSeq{1}{\nbsommets}}\do%
		{%
			\itemtomacro\readcaractdiv[\arbredivnbdivprem,2]\tmppuiss%
			\xdef\tmpdiv{\xinteval{\tmppuiss+1}}%
			\node[outer sep=0pt] (N-\arbredivnbdivprem-##1) at ({\arbredivnbdivprem*\TmpEspNiv},{-(##1-1)*\TMpEspFeuille}) {$\num{\readcaractdiv[\arbredivnbdivprem,1]}^{\xinteval{\xintiiRem{\xinteval{##1-1}}{\tmpdiv}}}$} ;%
		}
		% niveaux suivants, en descendant
		\xintFor* ##2 in {\xintSeq{\arbredivnbdivprem-1}{1}}\do%
		{%
			\xdef\nbsommets{\xintieval{\nbsommets/(\readcaractdiv[##2+1,2]+1)}}%
			\xdef\espacementsommets{\xinteval{\arbredivnbdiv/(\nbsommets)}}%
			\xdef\OffsetNiveau{\xintieval{(\HauteurTotale-(\nbsommets-1)*\espacementsommets)}}%
			\xintFor* ##3 in {\xintSeq{1}{\nbsommets}}\do%
			{%
				\itemtomacro\readcaractdiv[##2,2]\tmppuiss%
				\xdef\tmpdiv{\xinteval{\tmppuiss+1}}%
				\node[outer sep=0pt] (N-##2-##3) at ({##2*\TmpEspNiv},{(-(##3-1)*\espacementsommets-0.5*\OffsetNiveau)*\TMpEspFeuille}) {$\num{\readcaractdiv[##2,1]}^{\xinteval{\xintiiRem{\xinteval{##3-1}}{\tmpdiv}}}$} ;
			}%
		}%
		% racine
		\coordinate (Racine) at ({0},{-0.5*(\arbredivnbdiv-1)*(\TMpEspFeuille)}) ;
		% \draw (Racine) node {$\Omega$} ;
		% les flèches, qui partent de la racine
		\xintFor* ##4 in {\xintSeq{1}{\xinteval{\readcaractdiv[1,2]+1}}}\do%
		{%
			\draw[flechearbrediv] (Racine) -- (N-1-##4) ;
		}%
		% les flèches successives
		\foreach \nivdep in {1,...,\xinteval{\arbredivnbdivprem-1}}%
		{%
			\xdef\nbsommetsniv{1}%
			\foreach \i in {1,...,\nivdep}{\xdef\nbsommetsniv{\xinteval{\nbsommetsniv*(\readcaractdiv[\i,2]+1)}}}%calcul du nombre de sommets de départ
			% boucle sur sommets de départ
			\foreach \numsom in {1,...,\nbsommetsniv}%
			{%
				\xdef\nivplusun{\xinteval{\nivdep+1}}%
				\foreach \nbsousbranches in {1,...,\xinteval{\readcaractdiv[\nivplusun,2]+1}}%
				{%
					\xdef\sumsomarriv{\xinteval{ (\readcaractdiv[\nivplusun,2]+1)*(\numsom-1) + \nbsousbranches }}%
					\draw[flechearbrediv] (N-\nivdep-\numsom) -- (N-\nivplusun-\sumsomarriv) ;
				}
			}%
		}%
		\ifboolKV[arbrediviseurs]{Details}%
		{%
			% essai de génération des calculs := ouaiissssss
			\foreach \i in {1,...,\arbredivnbdiv}{%
				\xdef\resdiv{1}%
				% on extrait la premiere puissance := on obtient la répartition ?
				\itemtomacro\readcaractdiv[-1,2]\tmppuiss\xdef\tmpdiv{\xinteval{(\tmppuiss+1)}}%
				% création de la liste des puissances !!
				\xdef\calculdiviseur{\num{\readcaractdiv[-1,1]}^{\xinteval{\xintiiRem{\xinteval{\i-1}}{\tmpdiv}}}}%
				\xdef\resdiv{\xinteval{\resdiv*(\readcaractdiv[-1,1])^(\xinteval{\xintiiRem{\xinteval{\i-1}}{\tmpdiv}})}}%
				% on complète avec les autres parties ??
				\foreach \nbprem in {2,...,\arbredivnbdivprem}{%
					% il reste à stocker les produits des puissances, en descendant... erf.....
					\xdef\tmppuisscumul{1}%
					\foreach \k in {1,...,\xinteval{\nbprem-1}}{%
						\itemtomacro\readcaractdiv[-\k,2]\tmppuissrepet%
						\xdef\tmppuisscumul{\xinteval{\tmppuisscumul*(\tmppuissrepet+1)}}%
					}%
					\itemtomacro\readcaractdiv[-\nbprem,2]\tmppuiss%
					\xdef\tmpdivniv{\xinteval{(\tmppuiss+1)}}%
					% \xdef\tmpdiv{\xinteval{(\tmppuissrepet+1)}}%
					\xdef\calculdiviseur{\num{\readcaractdiv[-\nbprem,1]}^{\xinteval{\xintiiRem{\xintieval{trunc((\i-1)/(\tmppuisscumul),0)}}{\tmpdivniv}}}\times\calculdiviseur}%
					\xdef\resdiv{\xinteval{\resdiv*(\readcaractdiv[-\nbprem,1])^(\xinteval{\xintiiRem{\xintieval{trunc((\i-1)/(\tmppuisscumul),0)}}{\tmpdivniv}})}}%
				}%
				\draw ([xshift=1cm]N-\arbredivnbdivprem-\i) node[right,\TmpCoulDetails] {$\calculdiviseur=\num{\resdiv}$} ;
			}%
		}{}%
	\end{tikzpicture}%
}

%somme des chiffres
\ifthenelse{\isundefined{\SommeChiffres}}%
	{%
		\NewDocumentCommand\SommeChiffres{ m }{%
			\StrLen{#1}[\nbchiffres]%
			\xdef\tmpres{0}%
			\foreach \i in {1,...,\nbchiffres}{\StrChar{#1}{\i}[\tmpchf]\xdef\tmpres{\xinteval{\tmpres+\tmpchf}}}%
			\ensuremath{\num{\tmpres}}%
		}
	}%
	{%
		\RenewDocumentCommand\SommeChiffres{ m }{%
			\StrLen{#1}[\nbchiffres]%
			\xdef\tmpres{0}%
			\foreach \i in {1,...,\nbchiffres}{\StrChar{#1}{\i}[\tmpchf]\xdef\tmpres{\xinteval{\tmpres+\tmpchf}}}%
			\ensuremath{\num{\tmpres}}%
		}
	}%

%====CHIFFREMENTS !!
\xdef\aLPHaBeTMajusc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
\xdef\aLPHaBeTminusc{abcdefghijklmnopqrstuvwxyz}

\NewDocumentCommand\InverseModulo{ s m m }{%
	\xdef\PGCDD{\xinteval{gcd(#2,#3)}}%
	\xdef\resinvmod{#2{} n'est pas inversible modulo #3.}
		\xintFor* ##1 in {\xintSeq{1}{#3}}\do{\xintifboolexpr{\xintiiRem{\xinteval{#2*##1}}{#3} == 1}{\xdef\resinvmod{##1}}{}}%
	\IfBooleanTF{#1}%
		{%
			On a $\text{PGCD}(#2;#3)=\num{\PGCDD}$.~%
			\xintifboolexpr{\PGCDD != 1}%
				{Le PGCD étant différent de 1, on en déduit que #2 n'est pas inversible modulo #3.\relax}%
				{Le PGCD étant égal à 1, on en déduit que #2 admet un inverse modulo #3.\\De plus on a $#2 \times \resinvmod = \xinteval{#2*\resinvmod} \equiv \xintiirem{\xinteval{#2*\resinvmod}}{#3}\:[#3]$, donc \resinvmod\ est l'inverse de #2 modulo #3.\relax}%
		}%
		{}%
}

\defKV[chiffaffine]{a=\xdef\tmpcoeffa{#1},b=\xdef\tmpcoeffb{#1},modulo=\xdef\tmpmodulo{#1}}
\setKVdefault[chiffaffine]{a=3,b=12,Dechiffr=false,modulo=26}

\NewDocumentCommand\ChiffrementAffine{ O{} m }{%
	\restoreKV[chiffaffine]%
	\setKV[chiffaffine]{#1}%
	%pgcd stocké
	\xdef\PGCDD{\xinteval{gcd(\tmpcoeffa,\tmpmodulo)}}%
	\StrLen{#2}[\tmpnbcaract]%
	\ifboolKV[chiffaffine]{Dechiffr}%
	{%
		\xdef\resinvmod{0}%
		\xintFor* ##1 in {\xintSeq{1}{\tmpmodulo}}\do{
			\xintifboolexpr{\xintiiRem{\xinteval{(\tmpcoeffa)*(##1)}}{\tmpmodulo} == 1}%
			{\xdef\resinvmod{##1}}{}%
		}%
	}{}%
	\foreach \i in {1,...,\tmpnbcaract}{%
		\StrChar{#2}{\i}[\tmpchar]%
		\IfStrEq{\tmpchar}{ }%
			{~}%
			{%
				%majuscule minuscule
				\IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
					{%
						\StrPosition{\aLPHaBeTMajusc}{\tmpchar}[\tmpcoeffx]%
					}%
					{%
						\StrPosition{\aLPHaBeTminusc}{\tmpchar}[\tmpcoeffx]%
					}%
				\xdef\tmpcoeffx{\xinteval{\tmpcoeffx-1}}%
				\ifboolKV[chiffaffine]{Dechiffr}%
					{%
						\xintifboolexpr{\PGCDD == 1}%
							{%
								\xdef\tmpres{\xintiiRem{\xinteval{(\resinvmod)*(\tmpcoeffx)-(\resinvmod)*(\tmpcoeffb)}}{\tmpmodulo}}%
								\xdef\tmpres{\xinteval{\tmpres+1}}%
								\IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
									{%
										\StrChar{\aLPHaBeTMajusc}{\tmpres}%
									}%
									{%
										\StrChar{\aLPHaBeTminusc}{\tmpres}%
									}%
							}%
							{}%
					}%
					{%
						\xdef\tmpres{\xintiiRem{\xinteval{\tmpcoeffa*\tmpcoeffx+\tmpcoeffb}}{\tmpmodulo}}%
						\xdef\tmpres{\xinteval{\tmpres+1}}%
						\IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
							{%
								\StrChar{\aLPHaBeTMajusc}{\tmpres}%
							}%
							{%
								\StrChar{\aLPHaBeTminusc}{\tmpres}%
							}%
					}%
			}%
	}%
	\ifboolKV[chiffaffine]{Dechiffr}%
	{\xintifboolexpr{\PGCDD != 1}{Le message ne peut pas être déchiffré car $\text{PGCD}(\tmpcoeffa;\tmpmodulo)\neq1$ !\relax}{}}%
	{}%
}

\defKV[chiffhill]{Matrice=\xdef\tmpcoeffmat{#1},Modulo=\xdef\tmpmodulo{#1}}
\setKVdefault[chiffhill]{Matrice={1,2,3,5},Dechiffr=false,modulo=26}

\NewDocumentCommand\ChiffrementHill{ O{} m }{%
	\restoreKV[chiffhill]%
	\setKV[chiffhill]{#1}%
	%passage de la châine en nb pair
	\StrLen{#2}[\tmpnbcar]%
	\xintifboolexpr{\xintiirem{\tmpnbcar}{2} == 1}{\xdef\tmpchaine{#2A}}{\xdef\tmpchaine{#2}}%
	\StrLen{\tmpchaine}[\tmpnbblocs]%
	\xdef\tmpnbblocs{\xintieval{\tmpnbblocs/2}}%
	%extraction des coeffs de la matrice + déterminant
	\readlist*\coeffmathill{\tmpcoeffmat}%
	\itemtomacro\coeffmathill[1]{\tmpcoeffa}%
	\itemtomacro\coeffmathill[2]{\tmpcoeffb}%
	\itemtomacro\coeffmathill[3]{\tmpcoeffc}%
	\itemtomacro\coeffmathill[4]{\tmpcoeffd}%
	\xdef\detmathill{\xintieval{(\tmpcoeffa)*(\tmpcoeffd)-(\tmpcoeffb)*(\tmpcoeffc)}}%
	%chiffrement / déchiffrement
	\ifboolKV[chiffhill]{Dechiffr}%
	{%
		%le cas où la matrice n'est pas inversible
		\xintifboolexpr{\detmathill == 0}%
		{%
			La matrice $\begin{pmatrix} \tmpcoeffa & \tmpcoeffb \\ \tmpcoeffc & \tmpcoeffd \end{pmatrix}$ n'est pas inversible, donc pas de déchiffrement possible !\relax
		}%
		{%
			%inversibilité du déterminant
			\xdef\resinvmod{0}%
			\xintFor* ##1 in {\xintSeq{1}{\tmpmodulo}}\do{%
				\xintifboolexpr{\xintiiRem{\xinteval{(\detmathill)*(##1)}}{\tmpmodulo} == 1}%
				{\xdef\resinvmod{##1}}{}%
			}%
			\xintifboolexpr{\resinvmod == 0}%si det non inversible modulo
			{%
				Le déterminant de la matrice $\begin{pmatrix} \tmpcoeffa & \tmpcoeffb \\ \tmpcoeffc & \tmpcoeffd \end{pmatrix}$ (qui vaut $\detmathill$) n'est pas inversible modulo \tmpmodulo, donc pas de déchiffrement !\relax
			}%
			{%
				%on peut déchiffrer !!
				\foreach \i in {1,...,\tmpnbblocs}{%
					\xdef\tmpindicea{\xintieval{2*(\i-1)+1}}\xdef\tmpindiceb{\xintieval{2*(\i)}}%
					\StrChar{\tmpchaine}{\tmpindicea}[\tmpchara]%
					\IfSubStr{\aLPHaBeTMajusc}{\tmpchara}%
					{%
						\StrPosition{\aLPHaBeTMajusc}{\tmpchara}[\tmpcoeffx]%
					}%
					{%
						\StrPosition{\aLPHaBeTminusc}{\tmpchara}[\tmpcoeffx]%
					}%
					\xdef\tmpcoeffx{\xintieval{\tmpcoeffx-1}}%
					\StrChar{\tmpchaine}{\tmpindiceb}[\tmpcharb]%
					\IfSubStr{\aLPHaBeTMajusc}{\tmpcharb}%
					{%
						\StrPosition{\aLPHaBeTMajusc}{\tmpcharb}[\tmpcoeffy]%
					}%
					{%
						\StrPosition{\aLPHaBeTminusc}{\tmpcharb}[\tmpcoeffy]%
					}%
					\xdef\tmpcoeffy{\xintieval{\tmpcoeffy-1}}%
					%1ère lettre
					\xdef\tmpresa{\xintiiRem{\xintieval{(\resinvmod)*(\tmpcoeffd)*(\tmpcoeffx)-(\resinvmod)*(\tmpcoeffb)*(\tmpcoeffy)}}{\tmpmodulo}}%
					\xdef\tmpresa{\xinteval{\tmpresa+1}}%
					%2ème lettre
					\xdef\tmpresb{\xintiiRem{\xintieval{-(\resinvmod)*(\tmpcoeffc)*(\tmpcoeffx)+(\resinvmod)*(\tmpcoeffa)*(\tmpcoeffy)}}{\tmpmodulo}}%
					\xdef\tmpresb{\xinteval{\tmpresb+1}}%
					%affichage des deux caractères
					\IfSubStr{\aLPHaBeTMajusc}{\tmpchara}%
					{%
						\StrChar{\aLPHaBeTMajusc}{\tmpresa}%
					}%
					{%
						\StrChar{\aLPHaBeTminusc}{\tmpresa}%
					}%
					\IfSubStr{\aLPHaBeTMajusc}{\tmpcharb}%
					{%
						\StrChar{\aLPHaBeTMajusc}{\tmpresb}%
					}%
					{%
						\StrChar{\aLPHaBeTminusc}{\tmpresb}%
					}%
				}%
			}%
		}%
	}%
	{%
		\foreach \i in {1,...,\tmpnbblocs}{%
			\xdef\tmpindicea{\xintieval{2*(\i-1)+1}}\xdef\tmpindiceb{\xintieval{2*(\i)}}%
			\StrChar{\tmpchaine}{\tmpindicea}[\tmpchara]%
			\IfSubStr{\aLPHaBeTMajusc}{\tmpchara}%
			{%
				\StrPosition{\aLPHaBeTMajusc}{\tmpchara}[\tmpcoeffx]%
			}%
			{%
				\StrPosition{\aLPHaBeTminusc}{\tmpchara}[\tmpcoeffx]%
			}%
			\xdef\tmpcoeffx{\xintieval{\tmpcoeffx-1}}%
			\StrChar{\tmpchaine}{\tmpindiceb}[\tmpcharb]%
			\IfSubStr{\aLPHaBeTMajusc}{\tmpcharb}%
			{%
				\StrPosition{\aLPHaBeTMajusc}{\tmpcharb}[\tmpcoeffy]%
			}%
			{%
				\StrPosition{\aLPHaBeTminusc}{\tmpcharb}[\tmpcoeffy]%
			}%
			\xdef\tmpcoeffy{\xintieval{\tmpcoeffy-1}}%
			%1ère lettre
			\xdef\tmpresa{\xintiiRem{\xintieval{(\tmpcoeffa)*(\tmpcoeffx)+(\tmpcoeffb)*(\tmpcoeffy)}}{\tmpmodulo}}%
			\xdef\tmpresa{\xinteval{\tmpresa+1}}%
			%2ème lettre
			\xdef\tmpresb{\xintiiRem{\xintieval{(\tmpcoeffc)*(\tmpcoeffx)+(\tmpcoeffd)*(\tmpcoeffy)}}{\tmpmodulo}}%
			\xdef\tmpresb{\xinteval{\tmpresb+1}}%
			%affichage des deux caractères
			\IfSubStr{\aLPHaBeTMajusc}{\tmpchara}%
			{%
				\StrChar{\aLPHaBeTMajusc}{\tmpresa}%
			}%
			{%
				\StrChar{\aLPHaBeTminusc}{\tmpresa}%
			}%
			\IfSubStr{\aLPHaBeTMajusc}{\tmpcharb}%
			{%
				\StrChar{\aLPHaBeTMajusc}{\tmpresb}%
			}%
			{%
				\StrChar{\aLPHaBeTminusc}{\tmpresb}%
			}%
		}%
	}%
}

\defKV[chiffcesar]{Decal=\xdef\tmpdecalcesar{#1}}
\setKVdefault[chiffcesar]{Decal=5,Dechiffr=false}

\NewDocumentCommand\ChiffrementCesar{ O{} m }{%
	\restoreKV[chiffcesar]%
	\setKV[chiffcesar]{#1}%
	\StrLen{#2}[\tmpnbcaract]%
	\foreach \i in {1,...,\tmpnbcaract}{%
		\StrChar{#2}{\i}[\tmpchar]%
		%majuscule minuscule
		\IfStrEq{\tmpchar}{ }%
			{~}%
			{%
				\IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
					{%
						\StrPosition{\aLPHaBeTMajusc}{\tmpchar}[\tmpcoeffx]%
					}%
					{%
						\StrPosition{\aLPHaBeTminusc}{\tmpchar}[\tmpcoeffx]%
					}%
				\ifboolKV[chiffcesar]{Dechiffr}%
					{\xdef\tmpcoeffx{\xintiirem{\xinteval{\tmpcoeffx-\tmpdecalcesar}}{26}}}%
					{\xdef\tmpcoeffx{\xintiirem{\xinteval{\tmpcoeffx+\tmpdecalcesar}}{26}}}%
				\IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
					{%
						\StrChar{\aLPHaBeTMajusc}{\tmpcoeffx}%
					}%
					{%
						\StrChar{\aLPHaBeTminusc}{\tmpcoeffx}%
					}%
			}%
	}%
}

%====DIV EUCL (OK), 2 versions, mises à jour ;-)
\setKVdefault[diveucl]{%
	Quotient=true,%
	Reste=true,%
	Vide=false,%
	Pointilles=\ldots
}
\NewDocumentCommand\DivEucl{ s O{} m m }{%
	\restoreKV[diveucl]%
	\setKV[diveucl]{#2}%
	\ifboolKV[diveucl]{Vide}%
		{%
			\setKV[diveucl]{Quotient=false,Reste=false}%
		}%
		{}%
	\xdef\tmpAA{\xinteval{#3}}\xdef\tmpBB{\xinteval{#4}}%
	\xdef\tmpQuotient{\xintiiQuo{\tmpAA}{\tmpBB}}\xdef\tmpReste{\xintiiRem{\tmpAA}{\tmpBB}}%
	\ensuremath{\num{\xinteval{#3}}=\num{\xinteval{#4}}\times\ifboolKV[diveucl]{Quotient}{\xintifboolexpr{\tmpQuotient < 0}{(\num{\tmpQuotient})}{\num{\tmpQuotient}}}{\useKV[diveucl]{Pointilles}}+\ifboolKV[diveucl]{Reste}{\num{\tmpReste}}{\useKV[diveucl]{Pointilles}}}%
	\IfBooleanT{#1}{~avec $0 \leqslant \num{\xintiiRem{\tmpAA}{\tmpBB}} < \xintifboolexpr{\tmpBB < 0}{\lvert\num{\xinteval{#4}}\rvert}{\num{\xinteval{#4}}}$}%
}

\NewDocumentCommand\DivisionEucl{ m m }{%
	\xdef\tmpAA{\xinteval{#1}}\xdef\tmpBB{\xinteval{#2}}%
	\xdef\tmpQuotient{\xintiiQuo{\tmpAA}{\tmpBB}}\xdef\tmpReste{\xintiiRem{\tmpAA}{\tmpBB}}%
	\ensuremath{\num{\tmpBB}\times\xintifboolexpr{\tmpQuotient < 0}{(\num{\tmpQuotient})}{\num{\tmpQuotient}}+\num{\xintiiRem{\tmpAA}{\tmpBB}}}
}

%====ADDITIONS POSÉES ?
\newlength{\colspecbinadd}
\setlength{\colspecbinadd}{2pt}

\NewDocumentCommand\PosRetenue{ O{red} m }{%
	%\tmptextcircled[#1]{#2}%
	\textcolor{#1}{\scalebox{\addbinposescret}[\addbinposescret]{#2}}%
}

\defKV[AddBinaire]{%
	CouleurRetenue=\def\addbinposecol{#1},%
	Espacement=\setlength{\colspecbinadd}{#1},%
	Police=\def\addbinposefont{#1},%
	EchelleRetenue=\def\addbinposescret{#1},%
	Base=\def\addbinposebase{#1}
}

\setKVdefault[AddBinaire]{%
	CouleurRetenue=red,%
	Espacement=2pt,%
	Police=\normalsize\normalfont,%
	EchelleRetenue=0.4,%
	Base=dec,%
	Egal=true
}

\NewDocumentCommand\PoseAddition{ O{} m m }{%
	\restoreKV[AddBinaire]%
	\setKV[AddBinaire]{#1}%
	%le résultat
	\IfStrEq{\addbinposebase}{dec}%
		{\xdef\tmpresultat{\xinteval{#2+#3}}}%
		{}%
	\IfStrEq{\addbinposebase}{bin}%
		{\xdef\tmpresultat{\xintDecToBin{\xinteval{\xintBinToDec{#2}+\xintBinToDec{#3}}}}}%
		{}%
	\IfStrEq{\addbinposebase}{hex}%
		{\xdef\tmpresultat{\xintDecToHex{\xinteval{\xintHexToDec{#2}+\xintHexToDec{#3}}}}}%
		{}%
	%\xdef\tmpresultat{\xintDecToBin{\xinteval{\xintBinToDec{#2}+\xintBinToDec{#3}}}}%
	%les calculs
	\StrLen{#2}[\longueurA]\StrLen{#3}[\longueurB]\StrLen{\tmpresultat}[\longueurC]%
	\xdef\longueurMax{\xinteval{max(\longueurA,\longueurB,\longueurC)}}%\longueurMax
	%le tableau
	\xintifboolexpr{\longueurB > \longueurA}%
		{%
			\xdef\ArgLigneHaut{#3}\xdef\ArgLigneBas{#2}%
			\xdef\addposeehaut{#3}\xdef\addposeebas{#2}%
		}%
		{%
			\xdef\ArgLigneHaut{#2}\xdef\ArgLigneBas{#3}%
			\xdef\addposeehaut{#2}\xdef\addposeebas{#3}%
		}%
	%création de la ligne du haut
	\StrLen{\ArgLigneHaut}[\LgHaut]%
	\StrChar{\ArgLigneHaut}{\LgHaut}[\LigneBinAdd]%
	\foreach \i in {1,...,\inteval{\LgHaut-1}}%
		{%
			\xdef\tmpcoeff{\xinteval{\LgHaut-\i}}%
			\StrChar{\ArgLigneHaut}{\tmpcoeff}[\TmpHaut]%
			\xdef\LigneBinAdd{\TmpHaut & \LigneBinAdd}%
		}%
	\xintifboolexpr{\longueurMax > \LgHaut}%
		{%
			\foreach \i in {1,...,\inteval{\longueurMax-\LgHaut}}%
				{\xdef\LigneBinAdd{& \LigneBinAdd}}%
		}%
		{}%
	%création de la ligne du bas
	\StrLen{\ArgLigneBas}[\LgBas]%
	\StrChar{\ArgLigneBas}{\LgBas}[\LigneBinAddB]%
	\foreach \i in {1,...,\inteval{\LgBas-1}}%
		{%
			\xdef\tmpcoeff{\xinteval{\LgBas-\i}}%
			\StrChar{\ArgLigneBas}{\tmpcoeff}[\TmpBas]%
			\xdef\LigneBinAddB{\TmpBas & \LigneBinAddB}%
		}%
	\xintifboolexpr{\longueurMax > \LgBas}%
		{%
			\foreach \i in {1,...,\inteval{\longueurMax-\LgBas}}%
			{\xdef\LigneBinAddB{& \LigneBinAddB}}%
		}%
		{}%
	%création de la ligne du résultat
	\StrLen{\tmpresultat}[\LgRes]%
	\StrChar{\tmpresultat}{\LgRes}[\LigneBinRes]%
	\foreach \i in {1,...,\inteval{\LgRes-1}}%
		{%
			\xdef\tmpcoeff{\xinteval{\LgRes-\i}}%
			\StrChar{\tmpresultat}{\tmpcoeff}[\TmpRes]%
			\xdef\LigneBinRes{\TmpRes & \LigneBinRes}%
		}%
	%création des retenues...
	\xdef\ArgBasZeros{\ArgLigneBas}%
	\xintifboolexpr{\LgHaut > \LgBas}%
		{%
			\foreach \i in {1,...,\inteval{\LgHaut-\LgBas}}%
			{\xdef\ArgBasZeros{0\ArgBasZeros}}%
		}%
		{}%
	%\ArgBasZeros\par
	\xdef\listeretenues{0}%
	\xdef\longueurmax{\xinteval{max(\longueurA,\longueurB)}}%\longueurMax
	\foreach \i in {1,...,\longueurmax}%
		{%
			\readlist*\lstretenues{\listeretenues}%
			\xdef\tmpcoeff{\xinteval{\longueurmax-\i+1}}%
			\itemtomacro\lstretenues[-\i]{\TmPNbR}%
			\StrChar{\ArgLigneHaut}{\tmpcoeff}[\TmPNbH]%
			\StrChar{\ArgBasZeros}{\tmpcoeff}[\TmPNbB]%
			%cas de la base
			\IfStrEq{\addbinposebase}{dec}%
				{\xdef\tmpresultat{\xinteval{\TmPNbR+\TmPNbH+\TmPNbB}}}%
				{}%
			\IfStrEq{\addbinposebase}{bin}%
				{\xdef\tmpresultat{\xintDecToBin{\xinteval{\xintBinToDec{\TmPNbR}+\xintBinToDec{\TmPNbH}+\xintBinToDec{\TmPNbB}}}}}%
				{}%
			\IfStrEq{\addbinposebase}{hex}%
				{\xdef\tmpresultat{\xintDecToHex{\xinteval{\xintHexToDec{\TmPNbR}+\xintHexToDec{\TmPNbH}+\xintHexToDec{\TmPNbB}}}}}%
				{}%
			%suite
			%\xdef\tmpresultat{\xintDecToBin{\xinteval{\xintBinToDec{\TmPNbR}+\xintBinToDec{\TmPNbH}+\xintBinToDec{\TmPNbB}}}}%
			\StrLen{\tmpresultat}[\LgRetenue]%
			\xintifboolexpr{\LgRetenue == 1}%si on met une retenue...
				{%
					\xdef\listeretenues{0,\listeretenues}%
				}%
				{%
					\StrGobbleRight{\tmpresultat}{1}[\tmpretenue]%
					\xdef\listeretenues{\tmpretenue,\listeretenues}%
				}%
		}%
	%On compte le nombre de zéros dans la ligne des retenues
	\StrCount{\listeretenues}{0}[\nbretenues]%\listeretenues\par\nbretenues\par%
	\readlist*\lstretenues{\listeretenues}%
	%\showitems*{\lstretenues}\par
	\xdef\LigneBinAddRet{\xintifboolexpr{\lstretenues[-1] == 0}{}{\PosRetenue[\addbinposecol]{\lstretenues[-1]}}}%
	\foreach \i in {2,...,\longueurMax}%
		{%
			\xdef\LigneBinAddRet{\xintifboolexpr{\lstretenues[-\i] == 0}{}{\PosRetenue[\addbinposecol]{\lstretenues[-\i]}} & \LigneBinAddRet}%
		}%
	%et on passe au tableau
	\xdef\longueurcasesope{\xinteval{\longueurMax-1}}%
	\xdef\longueurtotale{\xinteval{\longueurMax+1}}%
	\setlength{\tabcolsep}{\colspecbinadd}%
	%\begin{tabular}{@{\hskip0.5\colspecbinadd}c@{\hskip0.5\colspecbinadd}*{\longueurcasesope}{c@{\hskip\colspecbinadd}}c@{\hskip0.5\colspecbinadd}}
	\begin{NiceTabular}{c*{\longueurMax}{c}}
		\xintifboolexpr{\nbretenues > \longueurmax}{}{\RowStyle[cell-space-limits=1pt]{\addbinposefont} & \LigneBinAddRet \\}
		\RowStyle[nb-rows=*,cell-space-limits=2pt]{\addbinposefont}
		&\LigneBinAdd \\
		+&\LigneBinAddB \\
		\hline 
		\ifboolKV[AddBinaire]{Egal}{=}{}&\LigneBinRes \\
	\end{NiceTabular}
	%\end{tabular}%
}

\ifluatex

\RequirePackage{lualinalg}

\defKV[addbinlua]{%
	CouleurRetenue=\def\addbinluacol{#1},%
	Espacement=\def\addbinluaspace{#1},%
	Police=\def\addbinluafonte{#1},%
	Base=\def\addbinluabase{#1},%
	EchelleRetenue=\def\addbinluascret{#1}
}

\setKVdefault[addbinlua]{%
	CouleurRetenue=red,%
	Espacement=2pt,%
	Police=\normalsize\normalfont,%
	Egal=true,%
	Base=dec,%
	EchelleRetenue=0.4
}

\NewDocumentCommand\SommeElementsMatriceBin{ m m m m O{\poseretenue} }{%
	%1=matrice
	%2=colonne
	%3=ligne début
	%4=ligne arrivée
	\xdef\tmpsomme{0}%
	\foreach \lig in {#3,...,#4}{%
		\xdef\tmpsomme{\xinteval{\tmpsomme+\xintBinToDec{\matrixGetElement{#1}{\lig}{#2}}}}%calcul intermédiaire en binaire
	}%
	\xdef#5{\xintDecToBin{\tmpsomme}}%
}

\NewDocumentCommand\SommeElementsMatriceDec{ m m m m O{\poseretenue} }{%
	%1=matrice
	%2=colonne
	%3=ligne début
	%4=ligne arrivée
	\xdef\tmpsomme{0}%
	\foreach \lig in {#3,...,#4}{%
		\xdef\tmpsomme{\xinteval{\tmpsomme+\matrixGetElement{#1}{\lig}{#2}}}%calcul intermédiaire en binaire
	}%
	\xdef#5{\tmpsomme}%
}

\NewDocumentCommand\PoseAdditionLua{ O{} m }{%
	\restoreKV[addbinlua]%
	\setKV[addbinlua]{#1}%
	\setsepchar{+}%
	\readlist*\addbinlstoper{#2}%
	\xdef\nboperandes{\addbinlstoperlen}%
	\xdef\nboperandesmat{\xinteval{\addbinlstoperlen+1}}%
	%le résultat du calcul en binaire
	\xdef\tmpres{0}%
	\foreach \i in {1,...,\addbinlstoperlen}{%
		\itemtomacro\addbinlstoper[\i]{\tmpinteroper}%
		\IfStrEq{\addbinluabase}{dec}%
			{\xdef\tmpres{\xinteval{\tmpres+\tmpinteroper}}}%
			{}%
		\IfStrEq{\addbinluabase}{bin}%
			{\xdef\tmpres{\xinteval{\tmpres+\xintBinToDec{\tmpinteroper}}}}%
			{}%
		%\xdef\tmpres{\xinteval{\tmpres+\xintBinToDec{\tmpinteroper}}}%
	}%
	\IfStrEq{\addbinluabase}{dec}%
		{\xdef\tmpresultat{\tmpres}}%
		{}%
	\IfStrEq{\addbinluabase}{bin}%
		{\xdef\tmpresultat{\xintDecToBin{\tmpres}}}%
		{}%
	%\xdef\tmpresultat{\xintDecToBin{\tmpres}}%\tmpresultat%
	\StrLen{\tmpresultat}[\nbtotalchiffresres]%\nbtotalchiffresres%
	\xdef\nblignesmatriceintertmp{\xinteval{1+\nboperandes}}%
	\xdef\nblignesmatricetmp{\xinteval{2+\nboperandes}}%
	\matrixNew{MATINTER}{\nblignesmatricetmp,\nbtotalchiffresres,'zero'}%
	%création des lignes intermédiaires
	\foreach \nblig [count=\xi] in {2,...,\nblignesmatriceintertmp}{%
		\itemtomacro\addbinlstoper[\xi]{\tmpinteroper}%on extrait l'opérande
		\StrLen{\tmpinteroper}[\nbchiiffres]%on extrait son nb de chiffres
		\xdef\tmpoffset{\xinteval{\nbtotalchiffresres-\nbchiiffres+1}}%on calcule la position du 1er chiffre
		\foreach \poschar [count=\xii] in {\tmpoffset,...,\nbtotalchiffresres}{%
			\StrChar{\tmpinteroper}{\xii}[\tmpcoeffaddbin]%
			\matrixSetElement{MATINTER}{\nblig}{\poschar}{\tmpcoeffaddbin}%
		}%
	}%
	%création de la ligne résultat
	\foreach \j in {1,...,\nbtotalchiffresres}{%
		\StrChar{\tmpresultat}{\j}[\tmpcoeffaddbin]%
		\matrixSetElement{MATINTER}{\nblignesmatricetmp}{\j}{\tmpcoeffaddbin}%
	}%
	%gestion des retenues, de la droite vers la gauche
	\foreach \nbcol in {\nbtotalchiffresres,...,2}{%
		\IfStrEq{\addbinluabase}{dec}%
			{\SommeElementsMatriceDec{MATINTER}{\nbcol}{1}{\nboperandesmat}[\poseretenue]}%
			{}%
		\IfStrEq{\addbinluabase}{bin}%
			{\SommeElementsMatriceBin{MATINTER}{\nbcol}{1}{\nboperandesmat}[\poseretenue]}%
			{}%
		%\SommeElementsMatriceBin{MATINTER}{\nbcol}{1}{\nboperandesmat}[\poseretenue]%
		\xdef\nbcolmun{\xinteval{\nbcol-1}}%
		\StrLen{\poseretenue}[\nbchiffresretenue]%
		\xintifboolexpr{\nbchiffresretenue > 1}%
			{\StrGobbleRight{\poseretenue}{1}[\RETENUE]}%
			{\xdef\RETENUE{\poseretenue}}%
		\matrixSetElement{MATINTER}{1}{\nbcolmun}{\RETENUE}%
	}%
	%ligne des retenues pour savoir si il y en a !
	\matrixSubmatrix{MATINTERRETENUE}{MATINTER}{1}{1}{1}{\nbtotalchiffresres}%
	%création de la ligne des retenues
	\xdef\LigneBinAddRet{}%
	\foreach \i in {1,...,\nbtotalchiffresres}{%
		\xintifboolexpr{\matrixGetElement{MATINTER}{1}{\i} == 0}%
			{\xdef\LigneBinAddRet{\LigneBinAddRet &}}%
			{\xdef\LigneBinAddRet{\LigneBinAddRet & \textcolor{\addbinluacol}{\scalebox{\addbinluascret}[\addbinluascret]{\matrixGetElement{MATINTER}{1}{\i}}}}}%
	}%
	%création des lignes intermédiaires
	\xdef\LignesOperande{}%
	\foreach \i in {1,...,\nboperandes}{%
		\xdef\xxi{\xinteval{\i+1}}%
		\itemtomacro\addbinlstoper[\i]{\tmpinteroper}%on extrait l'opérande
		\StrLen{\tmpinteroper}[\nbchiiffres]%on extrait son nb de chiffres
		\xdef\tmpoffset{\xinteval{\nbtotalchiffresres-\nbchiiffres}}%on calcule le nombre de cases vides
		\xintifboolexpr{\i == 1}{}{\xdef\LignesOperande{\LignesOperande + }}%
		\xintifboolexpr{\tmpoffset == 0}%
			{}%
			{
				\foreach \k in {1,...,\tmpoffset}{%
					\xdef\LignesOperande{\LignesOperande &}%
				}%
			}%
		\foreach \j in {\inteval{\tmpoffset+1},...,\nbtotalchiffresres}{%
			\xdef\LignesOperande{\LignesOperande & \matrixGetElement{MATINTER}{\xxi}{\j}}%
		}%
		\xdef\LignesOperande{\LignesOperande \\}%
	}%
	%et le tableau !!
	\setlength{\tabcolsep}{\addbinluaspace}%
	\begin{NiceTabular}{c*{\nbtotalchiffresres}{c}}
		\xintifboolexpr{\matrixNormOne{MATINTERRETENUE} == 0}{}{\RowStyle[cell-space-limits=1pt]{\addbinluafonte} \LigneBinAddRet \\}
		\RowStyle[nb-rows=*,cell-space-limits=2pt]{\addbinluafonte}
		\LignesOperande
		\hline
		\ifboolKV[addbinlua]{Egal}{=}{} \xintFor* ##1 in {\xintSeq{1}{\nbtotalchiffresres}}\do{& \matrixGetElement{MATINTER}{\nblignesmatricetmp}{##1}} \\
	\end{NiceTabular}%
	%\(\matrixPrint{MATINTER}\)
}

\fi

%====RESTE MODULO
\NewDocumentCommand\ResteMod{ s m m }{%
	%étoilé := version négative;#2=nb;#3=base
	\IfBooleanTF{#1}%
		{\num{\xintiieval{irem(#2,#3)-#3}}}%
		{\num{\xintiieval{irem(#2,#3)}}}%
}

%====OPÉRATIONS POSÉES 2/10/16 MULTICOMPILATEUR ?
\RequirePackage{calc}

\newlength{\widestcharwd}
\newlength{\charwd}
\newlength{\heightercharht}

\defKV[poseoperation]{%
	Base=\def\PoseOpeBase{#1},%
	LimiteCapac=\def\PoseOpeLimit{#1},%
	SymbDecal=\def\PoseOpeSymb{#1},%
	Offset=\def\PoseOpeOffset{#1},%
	CouleurRetenue=\def\PoseOpeColReten{#1}
}

\setKVdefault[poseoperation]{%
	Base=dec,%
	SymbDecal=.,%
	Interm=true,%
	Offset=6pt,%
	AffEgal=true,%
	LimiteCapac=0,%
	CouleurRetenue=red,%
	AffRetenues=true
}

\NewDocumentCommand\IntCalcMaxWidth{ m }{%
	\StrLen{#1}[\tmplen]%
	\setlength{\widestcharwd}{0pt}
	\xintFor* ##1 in {\xintSeq{1}{\tmplen}}\do{%
		\StrChar{#1}{##1}[\tmpchar]%
		\settowidth{\charwd}{\tmpchar}%
		\setlength{\widestcharwd}{\maxof{\widestcharwd}{\charwd}}%
		
		
	}%
}

\NewDocumentCommand\IntCreateBoxNumbers{ O{c} m }{%
	\IfStrEqCase{\PoseOpeBase}{%
		{dec}{\IntCalcMaxWidth{0123456789}}%
		{bin}{\IntCalcMaxWidth{01}}%
		{hex}{\IntCalcMaxWidth{0123456789ABCDEF}}%
	}%
	\addtolength{\widestcharwd}{\PoseOpeOffset/2}%
	\StrLen{#2}[\tmplennumber]%
	\xintFor* ##1 in {\xintSeq{1}{\tmplennumber}}\do{%
		\StrChar{#2}{##1}[\tmpchiff]%
		\makebox[\the\widestcharwd][#1]{\tmpchiff}%
	}%
}

\newcommand\AffRetenue[1]{%
	\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{#1}}%
}

\NewDocumentCommand\IntCreateBoxNumbersRetenue{ m }{%
	\IfStrEqCase{\PoseOpeBase}{%
		{dec}{\IntCalcMaxWidth{0123456789}}%
		{bin}{\IntCalcMaxWidth{01}}%
		{hex}{\IntCalcMaxWidth{0123456789ABCDEF}}%
	}%
	\addtolength{\widestcharwd}{\PoseOpeOffset/2}%
	\StrLen{#1}[\tmplennumber]%
	\xintFor* ##1 in {\xintSeq{1}{\tmplennumber}}\do{%
		\StrChar{#1}{##1}[\tmpchiff]%
		\IfEq{\tmpchiff}{X}%
			{\makebox[\the\widestcharwd][c]{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{}}}}%
			{\makebox[\the\widestcharwd][c]{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\tmpchiff}}}}%
	}%
}

\NewDocumentCommand\IntPoseAddition{ m m D<>{} }{%
	\StrLen{#1}[\nbchiffA]%
	%essai des retenues ??
	\xdef\ListeRetenues{X}%
	\xintFor* ##1 in {\xintSeq{1}{\nbchiffA}}\do{%
		\StrRight{#1}{##1}[\tmpinterA]%
		\StrRight{#2}{##1}[\tmpinterB]%
		\IfStrEqCase{\PoseOpeBase}{%
			{dec}{\xdef\rescalcrete{\xinteval{\tmpinterA+\tmpinterB}}}%
			{bin}{\xdef\rescalcrete{\xintDecToBin{\xinteval{\xintBinToDec{\tmpinterA}+\xintBinToDec{\tmpinterB}}}}}%
			{hex}{\xdef\rescalcrete{\xintDecToHex{\xinteval{\xintHexToDec{\tmpinterA}+\xintHexToDec{\tmpinterB}}}}}%
		}%
		\StrLen{\rescalcrete}[\tmpnbchiffresinter]%
		\xintifboolexpr{\tmpnbchiffresinter > ##1}%
			{%
				\StrGobbleRight{\rescalcrete}{##1}[\tmpretenue]%
				\xdef\ListeRetenues{\tmpretenue\ListeRetenues}%
			}%
			{%
				\xintifboolexpr{##1 == \nbchiffA}{}{\xdef\ListeRetenues{X\ListeRetenues}}%
			}%
	}%
	%suite
	\IfStrEqCase{\PoseOpeBase}{%
		{dec}{\xdef\rescalcbin{\xinteval{#1+#2}}}%
		{bin}{\xdef\rescalcbin{\xintDecToBin{\xinteval{\xintBinToDec{#1}+\xintBinToDec{#2}}}}}%
		{hex}{\xdef\rescalcbin{\xintDecToHex{\xinteval{\xintHexToDec{#1}+\xintHexToDec{#2}}}}}%
	}%
	\xintifboolexpr{\PoseOpeLimit > 0 }{\StrRight{\rescalcbin}{\PoseOpeLimit}[\rescalcbin]}{}%
	%\ensuremath{\begin{array}{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
	\ensuremath{\begin{NiceArray}[#3]{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
		\ifboolKV[poseoperation]{AffRetenues}{\RowStyle[cell-space-limits=0pt]{} & \IntCreateBoxNumbersRetenue{\ListeRetenues} \\[-0.5\heightercharht]}{}
		\RowStyle[nb-rows=*,cell-space-limits=1pt]{}  & \IntCreateBoxNumbers{#1} \\
		+ & \IntCreateBoxNumbers{#2} \\
		\hline
		\ifboolKV[poseoperation]{AffEgal}{=}{} & \IntCreateBoxNumbers{\rescalcbin} \\
	\end{NiceArray}}%
	%\end{array}}%
}

\NewDocumentCommand\IntPoseSoustraction{ m m D<>{} }{%
	\IfStrEqCase{\PoseOpeBase}{%
		{dec}{\xdef\rescalcbin{\xinteval{#1-#2}}}%
		{bin}{\xdef\rescalcbin{\xintDecToBin{\xinteval{\xintBinToDec{#1}-\xintBinToDec{#2}}}}}%
		{hex}{\xdef\rescalcbin{\xintDecToHex{\xinteval{\xintHexToDec{#1}-\xintHexToDec{#2}}}}}%
	}%
	%bourrage de zéros pour les retenues
	\StrLen{#1}[\lenA]\StrLen{#2}[\lenB]%
	\xdef\BwithZeros{#2}%
	\xintifboolexpr{ \lenA > \lenB }%
		{%
			\xintFor* ##1 in {\xintSeq{1}{\xinteval{\lenA-\lenB}}}\do{\xdef\BwithZeros{0\BwithZeros}}%
		}%
		{}%
	%suite
	\xintifboolexpr{\PoseOpeLimit > 0 }{\StrRight{\rescalcbin}{\PoseOpeLimit}[\rescalcbin]}{}%
	%\ensuremath{\begin{array}{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
	\ifboolKV[poseoperation]{AffRetenues}{%
	\ensuremath{\begin{NiceArray}[#3]{@{\,}r@{\hspace{\PoseOpeOffset/2}}r@{\,}}
		\RowStyle[nb-rows=*,cell-space-limits=1pt]{}
		& \IfStrEqCase{\PoseOpeBase}{%
			{dec}{\IntCalcMaxWidth{0123456789}}%
			{bin}{\IntCalcMaxWidth{01}}%
			{hex}{\IntCalcMaxWidth{0123456789ABCDEF}}%
		}%
		\addtolength{\widestcharwd}{\PoseOpeOffset/2}%
		\StrLen{#1}[\tmplennumber]%
		\xintFor* ##1 in {\xintSeq{1}{\tmplennumber}}\do{%
			\xdef\tmplenctr{\inteval{\tmplennumber-##1+1}}%
			\StrChar{#1}{##1}[\tmpchiff]%
			\StrRight{#1}{\tmplenctr}[\tmpA]%
			\StrRight{\BwithZeros}{\tmplenctr}[\tmpB]%
			\IfStrEqCase{\PoseOpeBase}{%
				{dec}{\xdef\rescalctmp{\xinteval{\tmpA-\tmpB}}}%
				{bin}{\xdef\rescalctmp{\xinteval{\xintBinToDec{\tmpA}-\xintBinToDec{\tmpB}}}}%
				{hex}{\xdef\rescalctmp{\xinteval{\xintHexToDec{\tmpA}-\xintHexToDec{\tmpB}}}}%
			}%
			\xintifboolexpr{ \rescalctmp >= 0 }%
				{\makebox[\the\widestcharwd][r]{\tmpchiff}}%
				%{\makebox[\the\widestcharwd][r]{${}_{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\text{1}}}}$\kern-0.1em\tmpchiff}}%
				{\makebox[\the\widestcharwd][r]{\raisebox{-0.5\height}{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\text{1}}}}\kern-0.1em\tmpchiff}}%
		} \\
		- & \IfStrEqCase{\PoseOpeBase}{%
			{dec}{\IntCalcMaxWidth{0123456789}}%
			{bin}{\IntCalcMaxWidth{01}}%
			{hex}{\IntCalcMaxWidth{0123456789ABCDEF}}%
		}%
		\addtolength{\widestcharwd}{\PoseOpeOffset/2}%
		\StrLen{#1}[\tmplennumber]%
		\StrChar{#2}{\lenB}[\tmplastB]%
		\xintFor* ##1 in {\xintSeq{1}{\xinteval{\tmplennumber-1}}}\do{%
			\xdef\tmplenctr{\inteval{\tmplennumber-##1}}%
			\StrChar{\BwithZeros}{##1}[\tmpchiff]%
			\StrRight{#1}{\tmplenctr}[\tmpA]%
			\StrRight{\BwithZeros}{\tmplenctr}[\tmpB]%
			\IfStrEqCase{\PoseOpeBase}{%
				{dec}{\xdef\rescalctmp{\xinteval{\tmpA-\tmpB}}}%
				{bin}{\xdef\rescalctmp{\xinteval{\xintBinToDec{\tmpA}-\xintBinToDec{\tmpB}}}}%
				{hex}{\xdef\rescalctmp{\xinteval{\xintHexToDec{\tmpA}-\xintHexToDec{\tmpB}}}}%
			}%
			\xintifboolexpr{ \rescalctmp >= 0 }%
				{\makebox[\the\widestcharwd][r]{\xintifboolexpr{ ##1 <= \lenA-\lenB }{}{\tmpchiff}}}%
				%{\makebox[\the\widestcharwd][r]{${}_{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\text{1}}}}$\xintifboolexpr{ ##1 <= \lenA-\lenB }{}{\kern-0.1em\tmpchiff}}}%
				{\makebox[\the\widestcharwd][r]{\raisebox{-0.5\height}{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\text{1}}}}\xintifboolexpr{ ##1 <= \lenA-\lenB }{\kern0.25em}{\kern-0.1em\tmpchiff}}}%
		}\makebox[\the\widestcharwd][r]{\tmplastB} \\
		
		%\IntCreateBoxNumbers{#2} \\
		\hline
		\ifboolKV[poseoperation]{AffEgal}{=}{} &\IntCreateBoxNumbers[r]{\rescalcbin} \\
	\end{NiceArray}}%
	}%
	{%
	\ensuremath{\begin{NiceArray}[#3]{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
		\RowStyle[nb-rows=*,cell-space-limits=1pt]{}
		& \IntCreateBoxNumbers{#1} \\
		- & \IntCreateBoxNumbers{#2} \\
		\hline
		\ifboolKV[poseoperation]{AffEgal}{=}{} & \IntCreateBoxNumbers{\rescalcbin} \\
	\end{NiceArray}}%
	}%
	%\end{array}}%
}

\NewDocumentCommand\IntPoseMultiplication{ m m D<>{} }{%
	\StrLen{#1}[\nbligninit]%
	\StrLen{#2}[\nbligninter]%
	\IfStrEqCase{\PoseOpeBase}{%
		{dec}{\xdef\rescalcbin{\xinteval{#1*#2}}}%
		{bin}{\xdef\rescalcbin{\xintDecToBin{\xinteval{\xintBinToDec{#1}*\xintBinToDec{#2}}}}}%
		{hex}{\xdef\rescalcbin{\xintDecToHex{\xinteval{\xintHexToDec{#1}*\xintHexToDec{#2}}}}}%
	}%
	\xintifboolexpr{\PoseOpeLimit > 0 }{\StrRight{\rescalcbin}{\PoseOpeLimit}[\rescalcbin]}{}%
	%\ensuremath{\begin{array}{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
	\ensuremath{\begin{NiceArray}[#3]{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
		\RowStyle[nb-rows=*,cell-space-limits=1pt]{}
		& \IntCreateBoxNumbers{#1} \\
		\times & \IntCreateBoxNumbers{#2} \\
		\ifboolKV[poseoperation]{Interm}%
			{%
				\hline
				\xintFor* ##1 in {\xintSeq{1}{\nbligninter}}\do{%
					\xintifboolexpr{##1 == 1}{}{+}&
					\xdef\tmpindice{\inteval{\nbligninter-##1+1}}\StrChar{#2}{\tmpindice}[\tmpchiff]\IfStrEqCase{\PoseOpeBase}{{dec}{\xdef\rescalcbininter{\xinteval{#1*\tmpchiff}}}{bin}{\xdef\rescalcbininter{\xintDecToBin{\xinteval{\xintBinToDec{#1}*\xintBinToDec{\tmpchiff}}}}}{hex}{\xdef\rescalcbininter{\xintDecToHex{\xinteval{\xintHexToDec{#1}*\xintHexToDec{\tmpchiff}}}}}}%
					\IfEq{\tmpchiff}{0}%
						{\xdef\rescalcbininter{}\xintFor* ##2 in {\xintSeq{1}{\nbligninit}}\do{\xdef\rescalcbininter{0\rescalcbininter}}%
						}{}%
					\xintifboolexpr{\PoseOpeLimit > 0 }{\StrRight{\rescalcbininter}{\PoseOpeLimit}[\rescalcbininter]}{}%
					\IntCreateBoxNumbers{\rescalcbininter}
					\xintifboolexpr{##1 == 1}{}{\xintFor* ##2 in {\xintSeq{1}{##1-1}}\do{\makebox[\the\widestcharwd][c]{\PoseOpeSymb}}}\\
				}%
			}%
			{}%
		\hline
		\ifboolKV[poseoperation]{AffEgal}{=}{} & \IntCreateBoxNumbers{\rescalcbin} \\
	\end{NiceArray}}%
	%\end{array}}%
}

\NewDocumentCommand\OperationPosee{ O{} m D<>{} }{%
	\restoreKV[poseoperation]%
	\setKV[poseoperation]{#1}%
	\IfStrEqCase{\PoseOpeBase}{%
		{dec}{\settoheight{\heightercharht}{\hbox{0123456789}}}%
		{bin}{\settoheight{\heightercharht}{\hbox{01}}}%
		{hex}{\settoheight{\heightercharht}{\hbox{0123456789ABCDEF}}}%
	}%
	\IfSubStr{#2}{+}%
		{%
			\StrCut{#2}{+}{\tmpcalcA}{\tmpcalcB}%
			\IntPoseAddition{\tmpcalcA}{\tmpcalcB}<#3>
		}%
		{}%
	\IfSubStr{#2}{-}%
		{%
			\StrCut{#2}{-}{\tmpcalcA}{\tmpcalcB}%
			\IntPoseSoustraction{\tmpcalcA}{\tmpcalcB}<#3>
		}%
		{}%
	\IfSubStr{#2}{*}%
		{%
			\StrCut{#2}{*}{\tmpcalcA}{\tmpcalcB}%
			\IntPoseMultiplication{\tmpcalcA}{\tmpcalcB}<#3>
		}%
		{}%
}

\endinput