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\begin{document}

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\noindent
\parbox{2.25in}{\bf Math 2110-410}
\hfill {\Large \bf Homework \#3} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Feb. $12^\mathrm{th}$, 2020}

\vspace{0.2in}

\begin{enumerate}
\item Use induction to show that $1+3+\cdots+(2n-1) = n^2$ for all integers $n \geq 1$.

         {\it Note:} Use \LaTeX\   to type up problem 1.

\item Prove that 8 is a factor of $ 9^n - 1 $ for all natural numbers $ n $.
\item Prove that $ n^2+n $ is always even [when $ n $ is an integer].

         {\it Note:} Use \LaTeX\   to type up problem 3.

\item Prove Theorem 20 (page 80): If no natural number $ m $ such that $ 1 < m \leq \sqrt{p} $ divides $ p $, then $ p $ is prime.
\item Prove Theorem 34 (page 82): The sum of any three consecutive integers is divisible by 3.
\end{enumerate}

Of course, you could type up all of your solutions, but I'm only requiring that you do this for problems 1. \& 3.


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