\documentclass{article}
\usepackage{amsmath, amssymb,verbatim, amsthm, graphicx}

%exercise environment
\newcounter{exercise}
\setcounter{exercise}{0}
\newenvironment{exercise}{\addtocounter{exercise}{1} \noindent{\bf{Exercise \theexercise.}}}{\vspace{.5cm}}

\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\mat}[4]{\left( \begin{array}{cc} #1 & #2 \\ #3 & #4  \end{array} \right)}
\newcommand{\smat}[4]{\left( \begin{smallmatrix} #1 & #2 \\ #3 & #4  \end{smallmatrix} \right)}

\textwidth 6.5in
\textheight 9in
\oddsidemargin 0in
\evensidemargin 0in
\headsep -0.5in

\setlength{\parskip}{0.25cm}

\pagenumbering{gobble}

\begin{document}

\begin{center}
\includegraphics[width=0.5in]{../ltr-logo.jpg}
\end{center}

\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 6.1  \\ Spring 2023 \hfill Due February 22}}}

\large

\vspace{1cm}

\begin{exercise}
Let $\ell, m, n \in \Z$ with $\ell | n$ and $m|n$.  Let $d = \gcd \left( \frac{n}{\ell}, \frac{n}{m} \right)$ and set $A = \langle \ell \rangle + \langle m \rangle$ in $\Z_n$.  

\begin{itemize}

\item[\bf a.] Show that $A = \langle \ell \rangle \oplus \langle m \rangle$ if and only if $d=1$.

\item[\bf b.] Show that $\Z_n =  \langle \ell \rangle \oplus \langle m \rangle$ if and only if $d=1$ and $n = \ell m $.

\end{itemize}

\end{exercise}

\begin{exercise}
Let $G$ be a group and $S \subseteq G$.  Given $x, y \in G$, define $x \sim y$ by the condition $xy^{-1} \in S$.  Prove that if $\sim$ is an equivalence relation on $G$, then $S < G$.
\end{exercise}

\end{document}
