\documentclass{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{enumerate}
\usepackage{ifthen}
\usepackage{ulem} % for strikethrough \sout{...} command
\usepackage{color}

\setlength{\unitlength}{0.1in}

\setlength{\oddsidemargin}{-0.25in}
\setlength{\evensidemargin}{-0.25in}
\setlength{\topmargin}{-0.25in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\topskip}{0.0in}
\setlength{\textheight}{9.75in}
\setlength{\textwidth}{7in}

\newcommand{\bdot}{\, {\scriptstyle{\bullet}} \,}
\newcommand{\B}[1]{{\bf {#1}}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\newcommand{\myspace}{\parbox[c][0.25in]{0.25in}{\hfill}}
\newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}}
\newcommand{\force}{{\noindent $\mbox{\;}$}}

\newcommand{\diagquotient}[2]{\displaystyle{{#1 \atop \;}\hspace*{-0.1in} \mbox{\put(0,0){\line(2,1){2}}} \hspace*{0.1in} {\; \atop #2}}}

\begin{document}

\pagestyle{empty}

\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Jan. $30^\mathrm{th}$, 2015}

\vspace{0.2in}

\begin{enumerate}
\item Workin' mod 15.
   \begin{enumerate}
   \item Find the additive inverse of each element in $\mathbb{Z}_{15}$. 
   
   \vspace{0.1in}
   
   \item Find the multiplicative inverse (it is exists) of each element in $\mathbb{Z}_{15}$.\\
             {}[If a multiplicative inverse fails to exist, write ``DNE'' (does not exist).]
             
   \vspace{0.1in}
   
   \item Compute \quad $2^{-3} \cdot (5-11) \cdot 14^{999}+13$ \qquad (mod $15$).          
             
   \end{enumerate}

   \vspace{0.2in}

\item The Euclidean Algorithm
   \begin{enumerate}
   \item Use the Euclidean Algorithm to find the greatest common divisor (gcd) of 1202 and 42.

   \vspace{0.1in}

   \item Use the (extended) Euclidean Algorithm to find the greatest common divisor of $a=303$ and $b=63$, say $d=\mathrm{gcd}(a,b)$. Then determine integers $x$ and $y$ such that $ax+by=d$.

   \vspace{0.1in}

   \item Use the (extended) Euclidean Algorithm to find $67^{-1}$ in $U(12345)$. 
   \end{enumerate}

   \vspace{0.2in}

\item {}[Gallian Chapter 0 \#8] Let $d=\mbox{gcd}(a,b)$. If $a=da'$ and $b=db'$, show that $\mbox{gcd}(a',b')=1$.

   \vspace{0.2in}

\item {}[Gallian Chapter 0 \#10] Let $d=\mbox{gcd}(a,b)$ and $\ell=\mbox{lcm}(a,b)$ (lcm $=$ least common multiple). 
          Show that if $c$ divides $a$ and $c$ divides $b$, then $c$ divides $d$. Also, show that if $c$ is a multiple of $a$ and
          $c$ is a multiple of $b$, then $c$ is a multiple of $\ell$. 
\end{enumerate}

\end{document}