\section{Bonus}

\problem{}
Find the inverse of 19 in $\mathbb{Z}_{23}$ \\
\hint{Recall the Euclidean Algorithm}


\begin{solution}
	17
\end{solution}
\vfill

\problem{}
Prove Lagrange's theorem:

$$
	a^p = a \text{ (mod p)}
$$

For positive integers $a, p$

\vfill

\problem{}
Let $a$ and $m$ be integers so that $a < m$. \\
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