\input{preamble}
\begin{document}

\begin{center}
  Fall 2021 -- Math 328K -- 55385
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\section*{Homework 2}
\begin{center}
  \itshape due Tuesday, September 14, 14:00
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\begin{problem}
  Show that, $3 \mid a^3 - a$ for every $a \in \bZ$.
\end{problem}

\begin{problem}
  The \emph{greatest integer function} or \emph{floor function} $\lfloor \cdot \rfloor \colon \bR \to \bZ$ is defined as follows: for every real number $x \in \bR$, $\lfloor x \rfloor$ is the greatest integer which is less or equal to $x$.

  Using the floor function and the basic arithmetic operations, find an explicit formula for the quotient and remainder of two integers. Prove your result.
\end{problem}

\begin{problem}
  Suppose that $a,b \in \bZ$ and $a \mid b$. Show that $a^n \mid b^n$ for every $n \in \bN$.
\end{problem}

\begin{problem}
  The following are meant to help you avoid common mistakes.
  \begin{enumerate}[(a)]
  \item Find integers $a$, $b$ and $c$ such that $a \mid bc$ but $a \nmid b$ and $a \nmid c$.
  \item Find integers $a$, $b$ and $c$ such that each pair of them has a common divisor greater than $1$, but that all three of them together do not have a common divisor greater than $1$.
  \end{enumerate}
\end{problem}

\begin{problem}
  Let $a$ and $b$ be integers. Recall that a \emph{pair of Bezout coefficients} for $a$ and $b$ is a pair of integers $m,n \in \bZ$ such that
  \[m a + n b = (a,b).\]
  Prove that, for any fixed pair of integers $a$ and $b$, there are infinitely many pairs of Bezout coefficients.
\end{problem}

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