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{\bf \centerline{Math 223 Exam 2;  April 2, 2004}}\medskip
\noindent{\bf Instructions for questions 1--4.} Answer the questions 
in the spaces provided. They are worth {\sl 5 points each}.
\medskip

\noindent{\bf 1.} Use the Euclidean algorithm to calculate the gcd 
$d$ of $80$ and $35$ and express
$d$ in the form $d=80x+35y$ for integers $x$ and $y$.\vskip 4cm

\noindent{\bf 2.} Determine the least natural number $N$ such that 
$2^N\equiv 1\pmod {25}$.
\vskip 4cm
\noindent{\bf 3.}  Find a solution of the three simultaneous congruences
$x\equiv 1\pmod 2$, $x\equiv 3\pmod 5$ and $x\equiv 2\pmod 7$. \vskip 4cm

\noindent{\bf 4.} Determine all the rational zeros of the polynomial 
$f(x)=x^3-3x^2+4$.

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{\bf Instructions for Questions 5--8} Give careful statements of the 
indicated definitions, principles or theorems in the spaces provided.
These questions are worth {\sl 5 points each}. Points will be 
deducted for unclear or imprecise
  (or of course incorrect) answers.\bigskip
\noindent{\bf 5.} Give a careful definition of a prime natural 
number, and state the fundamental
  theorem of arithmetic.
\vskip 3.8 cm
\noindent{\bf 6.} Let $n\geq 2$ be a natural number. Explain what is 
meant by a congruence class modulo $n$. How many such (distinct) 
congruence classes are there?
\vskip 3.8cm
\noindent{\bf 7.}  State Fermat's little theorem.\vskip 4cm


\noindent{\bf 8.} Recall the construction of the rational numbers 
from the integers
  (involving  the set $F$ of ordered pairs of integers $(a,b)$ with 
$b\neq 0$ under
  a suitable equivalence relation $\sim$).
Describe the appropriate equivalence relation $\sim$ on $F$ 
explicitly. How is the fraction $a/b$ defined explicitly
  in terms of $F$ and $\sim$?\vskip 3cm

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{\bf Instruction for Questions 9--12} Questions 9--12 are True/False 
questions worth {\sl 5 points each}.
Write your answer (T or F) in the space provided. No working is 
required.\bigskip
\noindent{\bf 9.} The only solutions $x\in \bf Z$  of the equation 
$x^2\equiv 1\pmod 8$ are $x\equiv 1\pmod 8$ and $x\equiv -1\pmod 8$.
\bigskip Answer:\vskip 3cm
\noindent{\bf 10.} Let $X=[8]=\{1,2,3,\ldots,8\}$. Consider the 
partition of $X$ into the sets
$\{1,3,5,7\}$,  $\{2,4,6,8\}$. Then the   equivalence relation $R$ on 
$X$ corresponding to this partition is given
by $xRy$ iff $x,y\in X$ and $x-y$ is even.  \bigskip Answer: \vskip 3cm
\noindent{\bf 11.}  The congruence $57 x\equiv 1 \pmod {403}$ has a 
unique solution.
\bigskip Answer: \vskip 3cm
\noindent{\bf 12.} The set ${\bf Z}_n$ of integers  modulo $n$ forms 
a group with binary operation $\circ$ given by
  $[a]\circ [b]=[ab]$. \bigskip Answer:
\vskip 3cm

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{\bf Instruction for Questions 13--15} Provide careful proofs  of the
  indicated statements in the spaces provided. Points will be deducted 
for unclear or
  imprecise explanation.\bigskip
\noindent{\bf 13.} {\sl (15 points)} Prove that there are infinitely 
many prime numbers.\vskip 6.5cm
\noindent{\bf 14.} {\sl (10 points)} Prove: if $a$, $b$, $c$ are integers such
  that $a$ and $b$ are relatively prime  and $a$ divides $bc$, then 
$a$ divides $c$ \vskip 6.5cm
\noindent{\bf  15.} {\sl (15 points)} Prove that the relation defined 
on ${\bf R}\times {\bf R}$ by $(x,y)\sim
(a,b)$ if $x^2+y^2=a^2+b^2$ is an equivalence relation.

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