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\begin{document}

\begin{center}
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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 1.1  \\ Spring 2019 \hfill Due January 23}}}

\large

\vspace{1cm}

\begin{exercise}
Use induction to prove that for any $n \in \N$,
\[
\sum_{j=1}^n j^2 = \frac{n(n+1)(2n+1)}{6}.
\]
\end{exercise}

\begin{exercise}
Given $n \in \N$ and $m \in \Z$, let $R(m)$ denote the remainder when $m$ is divided by $n$.  Prove that for all $a, b, c \in \Z$,
\[
R(a + R(b + c)) = R(R(a + b) + c) \,\,\, \text{ and } \,\,\, R(a R(bc)) = R(R(ab)c)
\]
\end{exercise}

\end{document}
