Skip to main content Contents Index Prev Up Next \(\newcommand{\longdivision}[2]{#1\big)\!\!\overline{\;#2}}
\newcommand{\mlongdivision}[2]{\longdivision{#1}{#2}}
\renewcommand{\emptyset}{\{\}}
\newcommand{\blanksp}{\underline{\hspace{.25in}}}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\cspace}{-}
\newcommand{\Ta}{\mathtt{a}}
\newcommand{\Tb}{\mathtt{b}}
\newcommand{\Tc}{\mathtt{c}}
\newcommand{\Td}{\mathtt{d}}
\newcommand{\Te}{\mathtt{e}}
\newcommand{\Tf}{\mathtt{f}}
\newcommand{\Tg}{\mathtt{g}}
\newcommand{\Th}{\mathtt{h}}
\newcommand{\Ti}{\mathtt{i}}
\newcommand{\Tj}{\mathtt{j}}
\newcommand{\Tk}{\mathtt{k}}
\newcommand{\Tl}{\mathtt{l}}
\newcommand{\Tm}{\mathtt{m}}
\newcommand{\Tn}{\mathtt{n}}
\newcommand{\To}{\mathtt{o}}
\newcommand{\Tp}{\mathtt{p}}
\newcommand{\Tq}{\mathtt{q}}
\newcommand{\Tr}{\mathtt{r}}
\newcommand{\Ts}{\mathtt{s}}
\newcommand{\Tt}{\mathtt{t}}
\newcommand{\Tu}{\mathtt{u}}
\newcommand{\Tv}{\mathtt{v}}
\newcommand{\Tw}{\mathtt{w}}
\newcommand{\Tx}{\mathtt{x}}
\newcommand{\Ty}{\mathtt{y}}
\newcommand{\Tz}{\mathtt{z}}
\newcommand{\So}{\Tf}
\newcommand{\Sno}{\Tg}
\newcommand{\Si}{\Th}
\newcommand{\Sni}{\Tj}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\W}{\mathbb{W}}
\newcommand{\ZZ}{\Z}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\RR}{\R}
\newcommand{\F}{\mathbb{F}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\abs}[1]{|#1|}
\newcommand{\fmod}{\bmod}
\newcommand{\fdiv}{\,\mathrm{div}\,}
\newcommand{\lcm}{\mathrm{lcm}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\nr}[1]{\##1}
\newcommand{\gexp}[3]{#1^{#2 #3}}
\newcommand{\gexpp}[3]{\displaystyle\left(#1\right)^{#2 #3}}
\newcommand{\glog}[3]{\log_{#1}^{#3}#2}
\newcommand{\sol}[1]{{\color{blue}\textit{#1}}}
\newcommand{\gro}[1]{{\color{gray}#1}}
\newcommand{\todo}[1]{{\color{purple}TO DO: #1}}
\newcommand{\fixme}[1]{{\color{red}FIX ME: #1}}
\newcommand{\checkme}[1]{{\color{green}CHECK ME: #1}}
\newcommand{\degre}{^\circ}
\newcommand{\vect}[1]{\overrightarrow{#1}}
\newcommand{\nix}{}
\newcommand{\cox}[1]{\fcolorbox[HTML]{000000}{#1}{\phantom{M}}}
\newcommand{\tox}[1]{\texttt{\##1} \amp \cox{#1}}
\newcommand{\ttx}[1]{\texttt{\##1}}
\newcommand{\mox}[1]{\mathtt{\##1}}
\newcommand{\xx}{\mathtt{\#}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Chapter 4 Greatest Common Divisors
Objectives
Determine whether an integer divides another integer.
Recognize the Euclidean algorithm.
Compute greatest common divisors.
Compute the cofactors from a special case of Bézout’s identity.
In the following we introduce another important algorithm, namely the Euclidean Algorithm (
Algorithm 4.17 ). This algorithm gives us a way to systematically determine the greatest common divisor of two natural numbers. Then, we show how to use the computations in the Euclidean Algorithm (
Algorithm 4.17 ) to determine the integers whose existence is guaranteed by Bézout’s identity (
Theorem 4.25 ).
