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\noindent \textbf{Math 314 - Spring 2019 \hfill
Name:}
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\noindent \textbf{Mission 3} \hfill Due February 21, 2019 \vspace{-4mm}\\
\small \textit{Cryptography succeeds when itâ€™s no longer the weakest link.}\\\vspace{1mm} \hfill--- Ron Rivest
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\section*{Guidelines}
\begin{itemize}
\item All work must be shown for full credit.
\item You can use CoCalc to help solve the problems. If you do, print out your code.
\item You may work with classmates, but be sure to turn in your own written solutions. Write down the name(s) of anyone who helps you.
\item Check one:\\
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\framebox(12,12){} I worked with the following classmate(s):
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\noindent\framebox(12,12){} I did not receive any help on this assignment.
\end{itemize}
\section{Graded Problems}
\begin{enumerate}[1.]
\item Use the Euclidean Algorithm to find the gcd of 217 and 1078.
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\item Use the Euclidean algorithm to find $x$ and $y$ so that $23x+77y=1$. What is $23^{-1} \pmod{77}$?
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\item Use modular exponentiation to compute $5^{268} \pmod{17}$. Make sure to show your steps.
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\item Let $F_n$ be the $n$-th Fibonacci number, where $F_1=1$, $F_2=1$, and for $i>2$ \[ F_i = F_{i-1}+F_{i-2}.\]
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\item What is $\gcd(F_9,F_8)$? How many steps of Euclid's algorithm are needed?
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\item For any $n>2$ what is $\gcd(F_n,F_{n-1})$? How many steps does it take? Prove your answer. (Induction may be helpful...)
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\section{Recommended Exercises}
\noindent These will not be graded but are recommended if you need more practice.
\begin{itemize}
\item Section 3.13: \# 1, 4, 18
\end{itemize}
\end{enumerate}
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