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\noindent{\Large{\textsc{Intro to Abstract Mathematics\hfill Assignment 8.1  \\ Spring 2020 \hfill Due April 1}}}

\large

\vspace{1cm}


\begin{exercise}
Let $m \in \N^+$.  Suppose $n \in \Z$ and $n<0$.  Show that there exist $q, r \in \Z$ so that
\[
n = qm + r \,\,\,\text{ and } \,\,\, 0 \le r < m.
\]
[{\em Suggestion:}  Apply the division algorithm to $-n$ or $-n-1$.]
\end{exercise}

\begin{exercise}
Let $m, n \in \N^+$.  Use the division algorithm to write $n = qm + r$ with $q, r \in \N$ and $0 \le r < m$.  Prove that $\gcd(m,n) = \gcd(r,m)$.\footnote{This is the basis of the {\em Euclidean algorithm}, the most efficient way known to compute greatest common divisors.}  [{\em Suggestion:}  Proceed as in Exercise 3.2.1.]
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