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\title{Math 214 -- Foundations of Mathematics\\
Homework 5\\
{\large{\bf Due noon Oct 5, 2012}}}
\date{}
\author{Your name}


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Solve the following problems.  Show all your work.  Every problem worths 4 points. 

\begin{enumerate}


\item Let $a, b, c, d\in \Z$ with $a, c\not=0$. Prove that if $a | b$ and $c | d$, then $ac | (ad+bc)$. 


\item Prove that for integers $n$ and $m$, $3 | nm$ if and only if $3 | m$ or $3 | n$.  (Hint: in one direction, you may want to use Division Theorem to write $m,n$ in some special form)


\item Find all primes, $p$, that can be written as $p = n^3 - 1$ for some $n \in \N$. (Make sure that you have found all of them.)

   
\item Prove or disprove: if $a$ and $b$ are odd integers, then $4 | (a-b)$ or $4 | (a+b)$. 

\item Prove that for any $n\in \Z$, $n^2$ cannot be of the form $3m-1$, where $m\in \Z$. 


\item Show that an positive integer is a multiple of $9$ if and only if the sum of all digits of the integer is a multiple of $9$. 


\item Find $\gcd(51, 288)$ and $m, n \in \Z$ such that $\gcd(51,288) = 51n + 288m$ (Show your
intermediate quotients and remainders).





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