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\begin{document}

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\noindent{\Large{\textsc{Number Theory \hfill Assignment 2.2  \\ Fall 2023 \hfill Due September 6}}}

\large

\vspace{1cm}

\begin{exercise}
For each of the following pairs of integers $a$ and $b$ use the Euclidean Algorithm and ``$q$ matrices'' to compute $(a,b)$ and express it as a linear combination of $a$ and $b$.

\begin{itemize}

\item[\bf a.] $a=24$, $b=138$

\item[\bf b.] $a=119$, $b=272$

\item[\bf c.] $a=1769$, $b=2378$

\end{itemize}

\end{exercise}

\begin{exercise}
Recall that the {\em Fibonacci sequence} $\{ f_n\}$ is defined by setting $f_0 = 0$, $f_1 = 1$ and $f_{n+2} = f_{n+1} + f_n$ for $n \ge 0$.  What happens if you apply the Euclidean Algorithm to compute $(f_n, f_{n+1})$?  More specifically, what is the GCD and what are the ``$q$ matrices''?
\end{exercise}

\begin{exercise}
Use the results of the preceding exercise to find $r,s \in \Z$ so that $rf_n + s f_{n+1} = (f_n, f_{n+1})$.
\end{exercise}


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