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\begin{document}

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\noindent{\Large{\textsc{Number Theory\hfill Assignment 5.1  \\ Fall 2020 \hfill Due September 30}}}

\large

\vspace{1cm}


\begin{exercise}
Textbook exercise 4.4.1(a)-(c).
\end{exercise}

\begin{exercise}
Let $m, n \in \N$.  If $m | n$, prove that the reduction map
\[
a+ n \Z \mapsto a + m \Z,
\]
from $\Z /n\Z$ to $\Z /m \Z$, is a surjective ring homomorphism. 
\end{exercise}

\begin{exercise}
Use the Euclidean Algorithm to compute the (multiplicative) inverse of $43+ n\Z$ for the following values of $n$.

\begin{itemize}

\item[\bf a.] $n=47$

\item[\bf b.]  $n=51$

\item[\bf c.] $n=1000$

\end{itemize}

\end{exercise}

\begin{exercise}  Use the preceding exercise to solve the following linear congruences.

\begin{itemize}

\item[\bf a.]  $43 x \equiv 9 \pmod{47}$

\item[\bf b.]  $43 x \equiv 3 \pmod{51}$

\item[\bf c.]  $43 x \equiv 500 \pmod{1000}$

\end{itemize}

\end{exercise}

\end{document}
