Added group theory parts

Advanced/Group Theory/parts/03 bonus.tex

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+\section{Bonus}
+
+\problem{}
+Find the inverse of 19 in $\mathbb{Z}/23$ \\
+\hint{Recall the Euclidian Algorithm}
+
+
+\begin{solution}
+	17
+\end{solution}
+\vfill
+
+\problem{}
+Prove Lagrange's theorem:
+
+$$
+	a^p = a \text{ (mod p)}
+$$
+
+\vfill
+
+\problem{}
+Show that $a$ has an inverse mod $m$ iff $\gcd(a, m) = 1$ \\
+
+\begin{solution}
+	Assume $a^\star$ is the inverse of $a \pmod{m}$. \\
+	Then $a^\star \times a \equiv 1 \pmod{m}$ \\
+
+	Therefore, $aa^\star - 1 = km$, and $aa^\star - km = 1$ \\
+	We know that $\gcd(a, m)$ divides $a$ and $m$, therefore $\gcd(a, m)$ must divide $1$. \\
+	$\gcd(a, m) = 1$ \\
+
+	Now, assume $\gcd(a, m) = 1$. \\
+	By the Extended Euclidean Algorithm, we can find $(u, v)$ that satisfy $au+mv=1$ \\
+	So, $au-1 = mv$. \\
+	$m$ divides $au-1$, so $au \equiv 1 \pmod{m}$ \\
+	$u$ is $a^\star$.
+\end{solution}
+
+\vfill
+
+
+\problem{}
+Show that for any integers $a, b, c$, \\
+$\gcd(ac + b, a) = \gcd(a, b)$\\
+
+\vfill