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\title{Math 412  Homework 6}
\date{Due date: Oct 25, 2010}
\author{your name}


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Solve the following problems. Please remember to use complete sentences
and good grammar. 



\begin{enumerate}
\item Use public key $(k, m)=(3, 2669)$ to decrypt $12151224147100230116$.    (hint:  use $00, 01, \ldots, 25$ to denote $a, b, \ldots, z$; and break the string into numbers of 4 digits.)

\item Solve the equation:  $128x\equiv 833\pmod{1001}$. 

\item Solve the following system of linear congruences $x\equiv 1\pmod{2}, x\equiv 2\pmod{3}, x\equiv 3\pmod{5}, x\equiv 4\pmod{7}$ and $x\equiv 5\pmod{11}$. 

\item Find an integer that leaves a remainder of $9$ when it is divided by either $10$ or $11$, but that is divisible by $13$. 

\item Solve the following system of linear congruences: $x\equiv 2\pmod{9}, x\equiv 8\pmod{15}, x\equiv 10\pmod{25}$. 

\item Show that the system of congruences $x\equiv a_1\pmod{m_1},  x\equiv a_2\pmod{m_2}$ has a solution if and only if $(m_1, m_2)| (a_1-a_2)$.  Show that when there is a solution, it is unique modulo $[m_1, m_2]$. 









\end{enumerate}

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