Cryptography edits

Advanced/Cryptography/parts/0 euclidean.tex

@@ -13,10 +13,20 @@ Find $\gcd(20, 14)$ by hand.
 
 \vfill
 
-\theorem{The Division Algorithm}
+\theorem{The Division Algorithm}<divalgo>
 Given two integers $a, b$, we can find two integers $q, r$, where $0 \leq r < b$ and $a = qb + r$. \par
 In other words, we can divide $a$ by $b$ to get $q$ remainder $r$.
 
+\begin{instructornote}
+	\ref{divalgo} looks scary on paper, but it's quite simple. \par
+	Doing a small example on the board (like $14 \div 3$) may be a good idea. \par
+
+	\vspace{2mm}
+
+	For those that are new to modular arithmetic, you may want to explain how remainders,
+	clock-face counting, division algorithm, and modular arithmetic are all the same.
+\end{instructornote}
+
 \theorem{}<gcd_abc>
 For any integers $a, b, c$, \par
 $\gcd(ac + b, a) = \gcd(a, b)$
@@ -64,7 +74,12 @@ Using the output of the Euclidean algorithm,
 	% gcd = 1
 	% u = 11; v = -175
 \end{itemize}
-This is called the \textit{extended Euclidean algorithm}.
+This is called the \textit{extended Euclidean algorithm}. \par
+\hint{
+	You don't need to fully solve the last part of this question. \\
+	Understand how you \textit{would} do it, then move on.
+	Don't spend too much time on arithmetic.
+}
 
 %For which numbers $c$ can we find a $(u, v)$ so that $541u + 34v = c$? \\
 %For every such $c$, what are $u$ and $v$?