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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Sept. $2^\mathrm{nd}$, 2020}

\vspace{0.2in}

\begin{enumerate}
\item Workin' mod 14.
   \begin{enumerate}
   \item Find the additive inverse and order of each element in $\mathbb{Z}_{14}$. 
   
   \vspace{0.1in}
   
   \item Find the multiplicative inverse or indicate ``DNE'' (does not exist) for each element in $\mathbb{Z}_{14}$.  If the multiplicative inverse exists, that element belongs to $U(14)$. In this case, find the order of that element (in $U(14)$).           
             
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   \item Compute \quad $5^{-2} \cdot (4-10) \cdot 13^{999}+11$ \qquad (mod $14$).          
             
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   \item Compute $A^{-1}$ given $A = \begin{bmatrix} 1 & 5 \\ 4 & 9 \end{bmatrix} \in \mathrm{GL}_2(\mathbb{Z}_{14})$.          
             
   \end{enumerate}

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\item The Euclidean Algorithm
   \begin{enumerate}
   \item Use the Euclidean Algorithm to find the greatest common divisor (gcd) of 1234 and 542.

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   \item Use the (extended) Euclidean Algorithm to find the greatest common divisor of $a=1001$ and $b=53$, say $d=\mathrm{gcd}(a,b)$. Then determine integers $x$ and $y$ such that $ax+by=d$.

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   \item Use the (extended) Euclidean Algorithm to find $9^{-1}$ in $U(1000)$. 
   \end{enumerate}

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\item Let $d=\mbox{gcd}(a,b)$. If $a=da'$ and $b=db'$, show that $\mbox{gcd}(a',b')=1$. [Of course, $a,a',b,b',d \in \mathbb{Z}$.]

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\item Show that for every $n \in \mathbb{Z}$ we have $n^3 \equiv n$ (mod $6$).
\end{enumerate}

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