这是我的乳胶副本。寻求任何解释。
\documentclass[11pt]{amsart}
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\begin{document}
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\noindent {\Large \textbf{Chapter 5}}
\bigskip
\begin{enumerate}\addtolength{\itemsep}{.5\baselineskip}
\medskip\item[52] \textbf{Use the Euclidean Algorithm to find $\gcd(219, 69)$.}
\medskip
By using the Euclidean Algorithm, we get the following:
\begin{equation*}
\openup\jot
\begin{aligned}[t]
& a = bq_{1} + r_{1}\\
& b = r_{1}q_{2} + r_{2}\\
& r_{1} = r_{2}q_{3} + r_{3}\\
& r_{2} = r_{3}q_{4} + r_{4}
\end{aligned}
\qquad\qquad
\begin{aligned}[t]
&219 = 69(3) + 12\\
&69 = 12(5) + 9\\
&12 = 9(1) + 3\\
&9 = 3(3) + 0
\end{aligned}
\end{equation*}
The last non-zero remainder is the greatest common divisor, and so $gcd(219, 69) = 3$
\item[53] \textbf{Find integers m and n so that $\gcd(219, 69) = 219m + 69n$.}
\medskip
\noindent We need to find the values for m and n when:
$3=219m +69n$
\medskip
The first step to solve this is to take the Euclidean Algorithm that we solved from above and in each step of it, solve for the remainder.
\begin{equation*}
\openup\jot
\begin{aligned}[t]
&219 = 69(3) + 12\\
&69 = 12(5) + 9\\
&12 = 9(1) + 3\\
&9 = 3(3) + 0
\end{aligned}
\qquad\qquad
\begin{aligned}[t]
&12 = 219 - 69(3)\\
&9 = 69 - 12(5) \\
&3 = 12 - 9(1) \\
&0 = 9 - 3(3)
\end{aligned}
\end{equation*}
\medskip
\noindent We begin the process by using the equation
\medskip
\begin{tabular}{ll}
$3 = 12 - 9(1)$ & We can also rewrite this as, \\
$3 = 12 + 9(-1)$ & We can now substitute for what 9 is equal to \\
$3 = 12 + (69-12(5))(-1)$ & We can rewrite this equation as \\
$3 = 12 + (69+12(-5))(-1)$ & We then can simplify \\
$3 = 12 + (69(-1)+12(5))$ & Which further simplifies to \\
$3 = 12(6) + 69(-1)$ & We can now substitute in for what 12 is equal to \\
$3 = (219 - 69(3)(6)) + 69(-1)$ & We can rewrite this equation as\\
$3 = (219 + 69(-3)(6)) + 69(-1)$ & We then can simplify \\
$3 = (219(6) + 69(-18)) + 69(-1)$ & Which further simplifies to \\
$3 = 219(6) + 69(-19)$ &
\end{tabular}
\medskip
\noindent By the above equation we have solved $\gcd(219, 69) = 219m + 69n$ when $m=6$ and $n=-19$.
\item[54] \textbf{Use the Euclidean Algorithm to find gcd(10245, 5357).}
By using the Euclidean Algorithm, we get the following:
\begin{center}
\begin{tabular}{rll}
$10245$ & $=5357(1)$ & + $4888$ \\
$5357$ & $=4888(1)$ & + $469$ \\
$4888$ & $=469(10)$ & + $198$ \\
$469$ & $=198(2)$ & + $73$\\
$198$ & $=73(2)$ & + $52$ \\
$73$ & $=52(1)$ & + $21$ \\
$52$ & $=21(2)$ & + $10$ \\
$21$ & $=10(2)$ & + $1$ \\
$10$ & $=1(10)$ & \\
\end{tabular}
\end{center}
The last non-zero remainder is the greatest common divisor, and so $\gcd(10245, 5357) = 1$
\item[55] \textbf{Find integers m and n so that $\gcd(10245, 5357) = 10245m + 5357n$.}
\noindent We need to find the values for m and n when:
$1=10245m + 5357n$
\medskip
\noindent The first step to solve this is to take the Euclidean Algorithm that we solved from above and in each step of it, solve for the remainder.
\begin{equation*}
\openup\jot
\begin{aligned}[t]
& 10245 = 5357(1) + 4888 \\
& 5357 =4888(1) + 469 \\
& 4888 =469(10) + 198 \\
& 469 =198(2) + 73\\
& 198 =73(2) + 52 \\
& 73 =52(1) + 21 \\
& 52 =21(2) + 10 \\
& 21 =10(2) + 1 \\
& 10 =1(10)
\end{aligned}
\qquad\qquad
\begin{aligned}[t]
& 4888 = 10245 +5357(-1) \\
& 469 =5357 +4888(-1) \\
& 198 =4888+469(-10) \\
& 73 =469+198(-2) \\
& 52 =198+73(-2) \\
& 21 =73 + 52(-1) \\
& 10 =52+21(-2) \\
& 1 =21+ 10(-2)
\end{aligned}
\end{equation*}
\noindent We begin the process by using the equation
\begin{tabular}{ll}
$1 = 21 + 10(-2)$ & We can now substitute for what 10 is equal to \\
$1 = 21 + (52+21(-2))(-2)$ & We then can simplify \\
$1 = 21 + (52(-2)+21(4))$ & Which further simplifies to \\
$1 = 21(5) + 52(-2)$ & We can now substitute in for what 21 is equal to \\
$1 = (73 + 52(-1)(5)) + 52(-2)$ & We then can simplify \\
$1 = (73(5) + 52(-5)) + 52(-2)$ & Which further simplifies to \\
$1 = 73(5) + 52(-7)$ & We can now substitute in for what 52 is equal to\\
$1 = 73(5) + (198+73(-2))(-7)$ & We then can simplify \\
$1 = 73(5) + 198(-7) +73(14)$ & Which further simplifies to \\
$1 = 73(19) + 198(-7) $ & We can now substitute in for what 73 is equal to \\
$1 = (469+198(-2))(19) + 198(-7) $ & We then can simplify \\
$1 = 469(19)+198(-38) + 198(-7) $ & Which further simplifies to \\
$1 = 469(19)+198(-45) $ & We can now substitute in for what 198 is equal to \\
$1 = 469(19)+(4888+469(-10))(-45) $ & We then can simplify \\
$1 = 469(19)+ 4888 (-45) +469(450) $ & Which further simplifies to \\
$1 = 469(469)+ 4888 (-45)$ & We can now substitute in for what 469 is equal to \\
$1 = (5357 +4888(-1))(469)+ 4888 (-45)$ & We then can simplify \\
$1 = 5357(469) +4888(-469)+ 4888 (-45)$ & Which further simplifies to \\
$1 = 5357(469) +4888(-514)$ & We can now substitute in for what 4888 is equal to \\
$1 = 5357(469) +(10245 +5357(-1))(-514)$ & We then can simplify \\
$1 = 5357(469) +10245(-514) +5357(514)$ & Which further simplifies to \\
$1 = 5357(983) +10245(-514) $ & \\
\end{tabular}
\medskip
\noindent By the above equation we have solved $\gcd(10245, 5357) = 10245m + 5357n$ when $m=-514$ and $n=983$.
\end{enumerate}
\end{document}
答案1
\medskip有时会产生不合理的大量垂直空白，原因是 (a)不是\medskip固定长度，而是所谓的“弹性”长度，并且 (b) 您有多个tabular无法跨页面拆分的大环境。因此，\medskip在各个地方会拉伸很多。
一种快速而（非常）粗暴的解决方案是\raggedbottom在 之后立即发出指令\begin{document}。这将停止实例的拉伸\medskip。但是，这种方法会在第 1 页和第 2 页的底部留下非常难看的间隙。
更好的解决方案是减少使用其他一些间距指令。例如，我认为您不需要在各种aligned环境中“打开”行距。通过这些调整，您的代码可以轻松适应约 1.8 页的输出。以下代码删除了语句\openup并执行了一些其他代码清理操作。我还缩短了最终tabular环境第二列中的短语，以确保它们tabular实际上适合文本块。
\documentclass[11pt]{amsart}
\usepackage{amsfonts}
\usepackage{graphics}
\usepackage{tikz}
\usepackage{booktabs}
%\usepackage{etoolbox}
%\makeatletter
%\preto{\@tabular}{\parskip=0pt}
%\makeatother
\usepackage{array} % <-- new
\newcolumntype{C}{>{{}}c<{{}}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newcommand{\abs}[1]{\left|#1\right|}
%\setlength{\textheight}{9in}
%\addtolength{\oddsidemargin}{-.875in}
%\addtolength{\evensidemargin}{-.625in}
%\addtolength{\textwidth}{1.50in}
%\addtolength{\textheight}{-.825in}
\usepackage[letterpaper,margin=1in]{geometry}
\setlength{\textfloatsep}{0pt}
\setlength{\belowdisplayskip}{0pt}
\setlength{\belowdisplayshortskip}{0pt}
\setlength{\abovedisplayskip}{0pt}
\setlength{\abovedisplayshortskip}{0pt}
\renewcommand\labelenumi{\textbf{\arabic{enumi}}}
\begin{document}
\pagestyle{empty}
\thispagestyle{empty}
{\Large\textbf{Chapter 5}}
\bigskip
\begin{enumerate}
\setcounter{enumi}{51}
\addtolength{\itemsep}{.5\baselineskip}
\item \textbf{Use the Euclidean Algorithm to find $\gcd(219, 69)$.}
\medskip\noindent
Using the Euclidean Algorithm, we get the following:
\begin{equation*}
%\openup\jot
\begin{aligned}[t]
a &= bq_{1} + r_{1}\\
b &= r_{1}q_{2} + r_{2}\\
r_{1} &= r_{2}q_{3} + r_{3}\\
r_{2} &= r_{3}q_{4} + r_{4}
\end{aligned}
\qquad\qquad
\begin{aligned}[t]
219 &= 69(3) + 12\\
69 &= 12(5) + 9\\
12 &= 9(1) + 3\\
9 &= 3(3) + 0
\end{aligned}
\end{equation*}
The last non-zero remainder is the greatest common divisor, and so $\gcd(219, 69) = 3$.
\item\textbf{Find integers $m$ and $n$ so that $\gcd(219, 69) = 219m + 69n$.}
\medskip\noindent
We need to find the values for $m$ and $n$ when:
\[3=219m +69n\]
The first step to solve this is to take the Euclidean Algorithm that we solved from above and in each step of it, solve for the remainder.
\begin{equation*}
%\openup\jot
\begin{aligned}[t]
219 &= 69(3) + 12\\
69 &= 12(5) + 9\\
12 &= 9(1) + 3\\
9 &= 3(3) + 0
\end{aligned}
\qquad\qquad
\begin{aligned}[t]
12 &= 219 - 69(3)\\
9 &= 69 - 12(5) \\
3 &= 12 - 9(1) \\
0 &= 9 - 3(3)
\end{aligned}
\end{equation*}
\medskip\noindent
We begin the process by using the equation
\[
\begin{tabular}{@{} >{$}l<{$} l @{}}
3 = 12 - 9(1) & We can also rewrite this as, \\
3 = 12 + 9(-1) & substitute for what 9 is equal to \\
3 = 12 + (69-12(5))(-1) & We can rewrite this equation as \\
3 = 12 + (69+12(-5))(-1) & We then can simplify \\
3 = 12 + (69(-1)+12(5)) & Which further simplifies to \\
3 = 12(6) + 69(-1) & substitute in for what 12 is equal to \\
3 = (219 - 69(3)(6)) + 69(-1) & We can rewrite this equation as\\
3 = (219 + 69(-3)(6)) + 69(-1) & We then can simplify \\
3 = (219(6) + 69(-18)) + 69(-1) & Which further simplifies to \\
3 = 219(6) + 69(-19) &
\end{tabular}
\]
By the above equation, we have solved $\gcd(219, 69) = 219m + 69n$ when $m=6$ and $n=-19$.
\item\textbf{Use the Euclidean Algorithm to find $\gcd(10245, 5357)$.}
\medskip\noindent
By using the Euclidean Algorithm, we get the following:
\[
\setlength\arraycolsep{0pt}
\begin{array}{rClCl}
10245 &=& 5357(1) &+& 4888 \\
5357 &=& 4888(1) &+& 469 \\
4888 &=& 469(10) &+& 198 \\
469 &=& 198(2) &+& 73\\
198 &=& 73(2) &+& 52 \\
73 &=& 52(1) &+& 21 \\
52 &=& 21(2) &+& 10 \\
21 &=& 10(2) &+& 1 \\
10 &=& 1(10) &
\end{array}
\]
The last non-zero remainder is the greatest common divisor, and so $\gcd(10245, 5357) = 1$.
\item\textbf{Find integers $m$ and $n$ so that $\gcd(10245, 5357) = 10245m + 5357n$.}
\medskip\noindent
We need to find the values for m and n when:
\[1=10245m + 5357n\]
The first step to solve this is to take the Euclidean Algorithm that we solved from above and in each step of it, solve for the remainder.
\begin{equation*}
\begin{aligned}[t]
10245 &= 5357(1) + 4888 \\
5357 &=4888(1) + 469 \\
4888 &=469(10) + 198 \\
469 &=198(2) + 73\\
198 &=73(2) + 52 \\
73 &=52(1) + 21 \\
52 &=21(2) + 10 \\
21 &=10(2) + 1 \\
10 &=1(10)
\end{aligned}
\qquad\qquad
\begin{aligned}[t]
4888 &= 10245 +5357(-1) \\
469 &=5357 +4888(-1) \\
198 &=4888+469(-10) \\
73 &=469+198(-2) \\
52 &=198+73(-2) \\
21 &=73 + 52(-1) \\
10 &=52+21(-2) \\
1 &=21+ 10(-2)
\end{aligned}
\end{equation*}
\noindent
We begin the process by using the equation
\[
\begin{tabular}{>{$}l<{$} l@{}}
1 = 21 + 10(-2) & Substitute for what 10 is equal to \\
1 = 21 + (52+21(-2))(-2) & We then can simplify \\
1 = 21 + (52(-2)+21(4)) & Which further simplifies to \\
1 = 21(5) + 52(-2) & Substitute in for what 21 is equal to \\
1 = (73 + 52(-1)(5)) + 52(-2) & We then can simplify \\
1 = (73(5) + 52(-5)) + 52(-2) & Which further simplifies to \\
1 = 73(5) + 52(-7) & Substitute in for what 52 is equal to\\
1 = 73(5) + (198+73(-2))(-7) & We then can simplify \\
1 = 73(5) + 198(-7) +73(14) & Which further simplifies to \\
1 = 73(19) + 198(-7) & Substitute in for what 73 is equal to \\
1 = (469+198(-2))(19) + 198(-7) & We then can simplify \\
1 = 469(19)+198(-38) + 198(-7) & Which further simplifies to \\
1 = 469(19)+198(-45) & Substitute in for what 198 is equal to \\
1 = 469(19)+(4888+469(-10))(-45) & We then can simplify \\
1 = 469(19)+ 4888 (-45) +469(450) & Which further simplifies to \\
1 = 469(469)+ 4888 (-45) & Substitute in for what 469 is equal to \\
1 = (5357 +4888(-1))(469)+ 4888 (-45) & We then can simplify \\
1 = 5357(469) +4888(-469)+ 4888 (-45) & Which further simplifies to \\
1 = 5357(469) +4888(-514) & Substitute in for what 4888 is equal to \\
1 = 5357(469) +(10245 +5357(-1))(-514) & We then can simplify \\
1 = 5357(469) +10245(-514) +5357(514) & Which further simplifies to \\
1 = 5357(983) +10245(-514) &
\end{tabular}
\]
By the above equation we have solved $\gcd(10245, 5357) = 10245m + 5357n$ for $m=-514$ and $n=983$.
\end{enumerate}
\end{document}