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\title{Math 480 (Spring 2007): Homework 1}
\author{\bf Due: Monday, April 2}
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\maketitle 

\noindent{\bf There are 7 problems.} Each problem is worth 5 points,
and parts of multi-part problems are worth equal amounts.\\
{\bf Office Hours.}
My official office hours are on Thursdays 4--6pm in Padelford C423.


\begin{enumerate}

\item  (This problem must be done without help from anyone else.)
Let $a,b,c,d$, and $m$ be integers.  Prove that
\begin{enumerate}
\item if $a\mid b$ and $b\mid c$ then $a\mid c$, %Jones&Jones
\item if $a\mid b$ and $c\mid d$ then $ac\mid bd$,
\item if $m\neq 0$, then $a\mid b$ if and only if $ma\mid mb$, and
\item if $d\mid a$ and $a\neq 0$, then $|d|\leq |a|$. 
\end{enumerate}

% pg 13 of kumanduri/romero
\item  (This problem must be done by hand
without help from anyone else.)
In each of the following, apply the division algorithm
to find $q$ and $r$ such that 
$a = bq + r$ and $0\leq r < |b|$:
$$
 a=300, b=17,\quad
a=729,b=31,\quad
a=300,b=-17,\quad
a=389,b=4.
$$

\item 
\begin{enumerate}
\item (Do this part  by hand.) Compute the greatest common
  divisor of $323$ and $437$ using the algorithm described in class
  that involves quotients and remainders (i.e., do not just factor $a$
  and $b$).
\item Compute by any means the greatest common divisor
$$
   \gcd(314159265358979323846264338, 271828182845904523536028747).
$$
\end{enumerate}

\item 
\begin{enumerate}
\item Suppose $a$, $b$ and $n$ are positive integers. Prove
that if $a^n\mid b^n$, then $a\mid b$.
\item Suppose $p$ is a prime and $a$ and $k$ are positive
integers.  Prove that if $p \mid a^k$, then $p^k \mid a^k$.
\end{enumerate}

%Burton, page 26
\item 
\begin{enumerate}
\item  Prove that if a positive integer $n$ is a perfect
square, then $n$ cannot be written in the form $4k+3$
for $k$ an integer.
(Hint: Compute the remainder upon division
by $4$ of each of $(4m)^2$, $(4m+1)^2$, $(4m+2)^2$,
and $(4m+3)^2$.)
\item  Prove that no integer in the sequence
$$
  11, 111, 1111, 11111, 111111, \ldots
$$
is a perfect square.   (Hint: $111\cdots111 = 111\cdots 108 + 3 = 4k+3$.)
\end{enumerate}

\item Prove that a positive integer $n$ is prime if
and only if $n$ is not divisible by any prime $p$
with $1 < p \leq \sqrt{n}$. 

\item So far $44$ Mersenne primes $2^p-1$ have been discovered.
Give a guess, backed up by an argument, about
when the next Mersenne prime might be discovered (you will have
to do some online research). 

\end{enumerate}
\end{document}




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