65 lines
1.6 KiB
TeX
Executable File
65 lines
1.6 KiB
TeX
Executable File
\section{The Euclidean Algorithm}
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\definition{}
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The \textit{greatest common divisor} of $a$ and $b$ is the greatest integer that divides both $a$ and $b$. \par
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We denote this number with $\gcd(a, b)$. For example, $\gcd(45, 60) = 15$.
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\problem{}
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Find $\gcd(20, 14)$ by hand.
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\begin{solution}
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$\gcd(20, 14) = 2$
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\end{solution}
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\vfill
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\theorem{The Division Algorithm}<divalgo>
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Given two integers $a, b$, we can find two integers $q, r$, where $0 \leq r < b$ and $a = qb + r$. \par
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In other words, we can divide $a$ by $b$ to get $q$ remainder $r$.
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\vspace{2mm}
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For example, take $14 \div 3$. We can re-write this as $3 \times 4 + 2$. \par
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Here, $a$ and $b$ are $14$ and $3$, $q = 4$ and $r = 2$.
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\theorem{}<gcd_abc>
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For any integers $a, b, c$, \par
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$\gcd(ac + b, a) = \gcd(a, b)$
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\problem{}
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Compute $\gcd(668, 6)$. \hint{$668 = 111 \times 6 + 2$}
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Then, compute $\gcd(3 \times 668 + 6, 668)$.
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\vfill
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\problem{The Euclidean Algorithm}
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Using the two theorems above, detail an algorithm for finding $\gcd(a, b)$. \par
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Then, compute $\gcd(1610, 207)$ by hand. \par
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\begin{solution}
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Using \ref{gcd_abc} and the division algorithm,
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% Minipage prevents column breaks inside body
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\begin{multicols}{2}
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\begin{minipage}{\columnwidth}
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$\gcd(1610, 207)$ \par
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$= \gcd(207, 161)$ \par
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$= \gcd(161, 46)$ \par
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$= \gcd(46, 23)$ \par
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$= \gcd(23, 0) = 23$ \par
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\end{minipage}
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\columnbreak
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\begin{minipage}{\columnwidth}
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$1610 = 207 \times 7 + 161$ \par
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$207 = 161 \times 1 + 46$ \par
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$161 = 46 \times 3 + 23$ \par
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$46 = 23 \times 2 + 0$ \par
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\end{minipage}
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\end{multicols}
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\end{solution}
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\vfill
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\pagebreak |