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\begin{center}
{\large\textbf{Dartmouth College}\\
Mathematics 25}
\end{center}
\begin{center}
Assignment 4\\
due Friday, October 23
\end{center}

\begin{enumerate}

\item Let $p$ be an odd prime, and $a$ an integer not divisible by
  $p$.  Show that $x^2 \equiv a \pmod{p}$ is solvable if and only if
  $x^2 \equiv a \pmod{p^2}$ is solvable.

\item Determine how many solutions there are to $x^2 \equiv 124
  \pmod{225}$ and find one of them using methods from the course.

\item Let $p$ be a prime and $u, v \in \Z$ with $u \equiv v
  \pmod{(p-1)}$.  Show that for any integer $a$, $a^u \equiv a^v
  \pmod{p}$.

\item  Let $p$ be an odd prime and $a$ an integer not divisible by
  $p$.  By Fermat's little theorem, we know that the set of positive
  integers $h$ so that $a^h \equiv 1 \pmod{p}$ is nonempty.  Denote
  the smallest such $h$ by $e_p(a)$.  Show that $e_p(a)$ divides $p-1$.
  Hint:  Whenever you want to show one integer divides another, you
  use the division algorithm and try to show the remainder is zero.

\item  Solve $3^{999} \equiv b\pmod{7}$ for $0\le b< 7$.

\item Find the least nonnegative residue of $7^{127} \pmod {12}$.

\item Show that for all integers $a$ with $\gcd(a,10) = 1$, that
$a^{20} \equiv 1\pmod{100}$.

\end{enumerate}

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