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{\bf Worksheet \#6; date: 09/17/2018}
{\bf MATH 55 Discrete Mathematics}
\begin{enumerate}
\item {\em (Rosen 4.1.7)} Show that if $a$, $b$, and $c$ are integers, where $a \ne 0$ and $c \ne 0$, such that $ac \mid bc$, then $a \mid b$.
\item {\em (Rosen 4.1.37a)} Find counterexample to the follow statement about congruences: If $ac \equiv bc \pmod{m}$, where $a$, $b$, $c$, and $m$ are integers with $m \ge 2$, then $a \equiv b \pmod{m}$.
\item {\em (Rosen 4.1.38)} Show that if $n$ is an integer then $n^2 \equiv 0 \text{ or } 1 \pmod{4}$.
\item {\em (Rosen 4.1.39)} Use the question above to show that if $m$ is a positive integer of the form $4k + 3$ for some nonnegative integer $k$, then $m$ is not the sum of the squares of two integers.
\item {\em (Rosen 4.2.23c)} Find the sum and product of the following pairs of numbers. Express your answers as an octal expansion.
\[
(1111)_8, (777)_8
\]
\item Show that a positive integer is divisible by $9$ if and only if the sum of its decimal digits is divisible by $9$.
\item {\em (Challenging)} Suppose we want to measure an item that weighs $n$ pounds, where $n$ is some unknown nonnegative integer no greater than $1000$. We are given a balance that has two sides to it, and we get to choose the mass of the weights used. Due to a budget crisis, we want to minimize the number of weights ordered. How do we achieve this, and how many weights are needed?
\end{enumerate}
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