\documentclass[12pt]{article}
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%% ------ Custom solution environment ---------
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  {\par\noindent\textbf{Solution.}\ } 
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%%----Custom Exercise environment-----
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{\end{exmp}}

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%% Put these after \begin{solution}, as necessary
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\pagestyle{fancy} \lhead{{\sf Homework 2 $|$ Due January 26 (Monday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-3mm}

%%------------------------------------------------------------------------

\begin{exercise} %% HW 2, Problem 1
Prove if $a,b\in\R$ and $ab$ is irrational, then at least one of $a$ or $b$ is irrational.
\end{exercise}

%\begin{solution} 
    %% Reminder to cite AI usage, if applicable. 
    % Put it immediately after \begin{solution} (on the same line)
    %\AIpolish
    %\AImajor
    %\AIminor
    %\AIpolishmajor
    %\AIpolishminor
    %
    % To do.
%\end{solution}

\medskip

%% You can add \smallskip, \medskip, \bigskip, or e.g., \vspace{5mm} if you want vertical spacing.


%%------------------------------------------------------------------------

\begin{exercise} %% HW 2, Problem 2
Let $n\in Z$. Prove that if $n^2$ is divisible by $3$, then $n$ is divisible by $3$. 
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 2, Problem 3
    Prove that $\sqrt{2}+\sqrt{3}$ is irrational. [\emph{Hint}: Suppose it were. Try solving for $\sqrt{3}$.]
\end{exercise}

%\begin{solution} %% Reminder to cite AI usage here, if applicable. 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------


\begin{exercise}% HW 2, Problem 4
 We proved the statement ``\emph{if $n^2$ is even, then $n$ is even}'' using the contrapositive. Alternatively, write a proof of this by contradiction.
\end{exercise}

%\begin{solution} %% Reminder to cite AI usage here, if applicable. 
    % To do.
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\end{document}
