\documentclass[12pt, letterpaper]{amsart}

\usepackage{amsmath, amssymb, amsthm}
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\usepackage[pdflang={en-US}]{hyperref}

\theoremstyle{definition}
\newtheorem{dfn}{Definition}
\newtheorem{problem}{}
\newtheorem*{instructions}{Instructions}

\title{Portfolio proofs A}
\date{\today}

\begin{document} \ 

\vspace{-.25in}

\maketitle

\vspace{-.2in} 

\begin{instructions}
Choose one of the following statements and prove it.
Use \LaTeX\ to write your proof nicely.
Drop your proof (both pdf and tex) in your OneDrive folder by the end of the day Wednesday, October 13.
\end{instructions}

\begin{problem}
Let $x \in \mathbb{R}$.
If $x>0$, then $\displaystyle x + \frac{1}{x} \geq 2$.
\end{problem}


\begin{problem}
Suppose $a \in \mathbb{Z}$.
If $a$ is odd, then $8 \mid (a^2 -1)$.
\end{problem}


\begin{problem}
Let $a, b, c, \in \mathbb{Z}$. 
Suppose $a$ and $b$ are not both zero and $c \neq 0$.
Prove that $c \cdot \gcd(a,b) \leq \gcd(ac, bc)$.
\end{problem}

% \begin{problem}
% Prove that if $n \in \mathbb{N}$ and $n \geq 2$, then the numbers $n! + 2$, $n! + 3$, $n! + 4$, \dots, $n! + n$ are all composite.
% (This means that $n! + 2$, $n! + 3$, $n! + 4$, \dots, $n! + n$ is a sequence of $n-1$ consecutive composite numbers, thus showing that there are arbitrarily large gaps between prime numbers).
% \end{problem}

\end{document}