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\centerline{Mathematics 243, section 1 -- Algebraic Structures}
\centerline{Makeup Exam 2, November 3, 2006}
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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 
$$(x_1,y_1)R(x_2,y_2) \Leftrightarrow x_1 - 3y_1 = x_2 - 3y_2$$
\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 $[(2,3)]$ for 
this relation.
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II. (20) Prove by mathematical induction:  For all $n \ge 1$,
$$1^2 + 3^2 + 5^2 + \cdots + (2n-1)^2 = {n(2n-1)(2n+1)\over 3}.$$
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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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IV.  (10)  Let $a,b,c$ be integers. 
Show that if $\gcd(a,b) = 1$ and $a|(bc)$, then $a|c$.  
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V.  
\item{A)} (10) Find a solution $x$ of the congruence 
$17 x \equiv 5 \bmod 32$ with $0 \le x < 31$.
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\item{B)} (10) For which $[b] \in {\bf Z}_{15}$ do solutions 
$[x] \in {\bf Z}_{15}$ of the equation $[6][x] = [b]$ exist?  
Explain how you can tell.
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