\documentclass[12pt,reqno]{amsart} 

\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{graphicx}
\usepackage{enumerate}
\usepackage[top=1in,bottom=1in,left=1in,right=1in]{geometry}
\usepackage{circuitikz}

\renewcommand{\baselinestretch}{1.25}

\newtheorem{thm}{Theorem}
\newtheorem{ex}{Exercise}

\begin{document}

\begin{flushright}
Your Name Here\\
Math 220\\
HW \# 5\\
\today
\end{flushright}

%Please write your solutions in the designated solution area!

\begin{ex}\noindent
\begin{enumerate}[$(a)$]
\item Is $52$ divisible by $13$?
\item Does $5|0$?
\item Is $(3k+1)(3k+2)(3k+3)$ divisible by $3$?
\item Is $29$ a multiple of $3$?
\item Is $-3$ a factor of $66$?
\item If $n=4k+1$, does $8$ divide $n^2-1$?
\end{enumerate}
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Let $a$ and $b$ be integers, with $a\neq 0$. Prove that if $a|b$, then $a^2|b^2$.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $a\in\mathbb{Z}$. Prove that if $3|2a$, then $3|a$.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $x\in\mathbb{Z}$. Prove that if $2|(x^2-5)$, then $4|(x^2-5)$.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $a,b,c\in\mathbb{Z}$ and assume $a^2+b^2=c^2$. Then $3|ab$.\\
\textit{Hint:} Use proof by contradiction. You will need some results from class to do this as well.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $a,b,n\in\mathbb{Z}$ with $n\geq 2$. Prove that if $a\equiv b\mod n$, then $a^2\equiv b^2\mod n$.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $a,b,c,n\in\mathbb{Z}$ with $n\geq 2$. Prove that if $a\equiv b\mod n$ and $a\equiv c\mod n$, then $b\equiv c\mod n$.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $m,n\in\mathbb{N}$ such that $m|n$. Prove that if $a$ and $b$ are integers such that $a\equiv b\mod n$, then $a\equiv b\mod m$.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Let $a\in\mathbb{Z}$. Prove that $a^3\equiv a\mod 3$.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
The product of any three consecutive integers is divisible by $6$.
\end{ex}

\begin{proof}

\end{proof}

\end{document}