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\centerline{Mathematics 243, section 2 -- Algebraic Structures}
\centerline{Makeup Exam 2, November 5, 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 
$$(x_1,y_1)R(x_2,y_2) \Leftrightarrow 3x_1 + y_1 = 3x_2 + y_2$$
\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 $n \ge 1$,
$$1^3 + 3^3 + 5^3 + \cdots + (2n-1)^3 = n^2(2n^2-1).$$
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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
$d = \gcd(576,99)$ and express $d$ in the form 
$d = m\cdot 576 + n\cdot 99$
for some integers $m,n$.
\item{C)} (15) Find all solutions $x \in {\bf Z}$ of the congruence 
$27 x \equiv 2 \bmod 64$.
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IV.  (15)  Let $a,b,c$ be integers. 
Show that if $\gcd(a,b) = 1$ and $\gcd(a,c) = 1$, then $\gcd(a,bc) = 1$.  
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{\it Extra Credit} (10)  Let $a_1 = 0$, $a_2 = -30$ and $a_n = 8a_{n-1} - 15a_{n-2}$
for all $n \ge 2$.  Show that $a_n = 5\cdot 3^n - 3\cdot 5^n$ for all $n\ge 1$.  
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