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{\bf Math 3140 - Assignment 2}
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Due September 7, 2016
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\begin{enumerate}
% \item
%  Find the order of the following permutations: \\
%  $a =  (1\ 3\ 5\ 7\ 9\ 11) (2\ 4\ 6\ 8) (10\ 12)$ \\ \smallskip
%  $b = \left(\begin{array}{ccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 7 & 3 & 1 & 6 & 5 & 4 & 2 \end{array}\right)$

\item
 Find the cycle decomposition for the perfect in-shuffle for $4,6,12$ cards, respectively.
 What are their orders?

\item
 Let $n\in\mathbb{N}, a,a',b,b'\in\mathbb{Z}$ such that $a \equiv_n a', b\equiv_n b'$.
 Show
 $$a+b \equiv_n a'+b',$$
\[ a\cdot b \equiv_n a'\cdot b'. \]

\item
 Use the Euclidean algorithm to find $\gcd(a,b)$ and Bezout's coefficients $u,v\in\Z$ such that 
\[ u\cdot a+ v\cdot b = \gcd(a,b) \]
 for
\begin{enumerate}
\item $a = 29, b = 13$, 
\item $a = 51, b = 36$.
\end{enumerate}

\item
 Compute the following multiplicative inverses in $\mathbb{Z}_n$ if possible:
\begin{enumerate}
\item $[12]^{-1}$ in $\Z_{35}$
\item $[14]^{-1}$ in $\Z_{35}$
\end{enumerate}
 Hint: Use the Euclidean Algorithm to compute Bezout's coefficients.

\item
 Let $n\in\N, n>1$. We call $[a]\in\Z_n$ a \emph{zero-divisor} if there exists some nonzero $[b]\in\Z_n$
 such that $[a]\cdot [b] = [0]$.

\begin{enumerate}
\item
 What are the zero divisors in $\Z_6$?
\item
 Show that $[a]$ is a zero divisor in $\Z_n$ if $\gcd(a,n) \neq 1$.
\item
 True or false? Every element in $\Z_n$ is either invertible or a zero-divisor.
\end{enumerate}

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% \begin{thebibliography}{1}
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%  Frederick M. Goodman. 
% \newblock Algebra: abstract and concrete.
% \newblock SemiSimple Press, edition 2.6, 2015.
% \newblock Available from \verb+http://www.math.uiowa.edu/~goodman+
% \end{thebibliography}



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