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\centerline{Mathematics 243, section 3 -- Algebraic Structures}
\centerline{Exam 2 -- November 2, 2012}
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I.  Think of ${\bf R}^2$ (the set of ordered pairs of real
numbers) as the ordinary Cartesian coordinate plane.
Let $R$ be the relation on ${\bf R}^2$ defined by 
$$(x_1,y_1)R(x_2,y_2) \Leftrightarrow x_1^2 + y_1^2 = x_2^2 + y_2^2$$
\item{A)} (15) Show that $R$ is an equivalence relation on ${\bf R}^2$.
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\item{B)} (5) Draw a picture of the equivalence class $[(3,4)]$ for 
this relation.
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II. (20) Prove by mathematical induction:  For all $n \ge 1$,
$${1\over 1\cdot 2\cdot 3} + {1\over 2\cdot 3\cdot 4} + \cdots + {1\over n(n+1)(n+2)} = 
  {n(n+3)\over 4(n+1)(n+2)}.$$
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III.  
\item{A)} (15) Give the precise statement of the division algorithm
in ${\bf Z}$, and prove the existence part. 
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\item{B)} (15) Use the Euclidean algorithm to find the integer
$d = \gcd(456,120)$ and express $d$ in the form 
$d = m\cdot 456 + n\cdot 120$
for some integers $m,n$.
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\item{C)}  (15) Find all solutions $x \in {\bf Z}$ of the congruence 
$17 x \equiv 5 \bmod 32$.
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IV.  (15)  Let $a,b,c$ be integers. 
Show that if $\gcd(a,b) = 1$ and $a|(bc)$, then $a|c$.  
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{\it Extra Credit} (10) Give a proof that every positive integer $n > 1$
is a product of prime numbers using complete induction.  (Note:  a ``product''
here may consist of a single factor.)
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