\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 B}
\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.
\end{instructions}

\begin{problem}
Let $n \in \mathbb{N}$. If $n \geq 2$, then $\displaystyle \sqrt[n]{2}$ is irrational. 
\end{problem}

\begin{problem}
If $a \in \mathbb{Z}$, then $4 \nmid (a^2 -3)$.
\end{problem}


\begin{problem}
For any natural numbers $a$ and $b$, $a = \text{lcm}(a,b)$ if and only if $b \mid a$.
\end{problem}

\begin{problem}
Let $C$ be a circle in $\mathbb{R}^2$ centered at $(1,1)$.
Then either $(2,3) \notin C$ or $(0, 2) \notin C$.
(Note that $C$ is just the circle itself, not the interior).
\end{problem}

\end{document}