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\begin{document}

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\noindent{\Large{\textsc{Number Theory\hfill Assignment 2.2  \\ Fall 2020 \hfill Due September 9}}}

\large

\vspace{1cm}


\begin{exercise}
Let $a, b \in \Z$, not both 0.  Prove that if $x, y \in \Z$ satisfy $ax+by = (a,b)$, then $(x,y) = 1$.  
\end{exercise}

\begin{exercise}
Show that for any integer $a \in \Z$, $(2a+1, 9a+4)=1$.
\end{exercise}

\begin{exercise}
If $a, b \in Z$ are not both zero, prove that $(2a-3b, 4a-5b)$ divides $b$.  Conclude that $(2a+3,4a+5)=1$.
\end{exercise}

\begin{exercise}
If $a, b, n \in \N$, prove that $(a,b)=1$ if and only if $(a^n,b^n) = 1$.
\end{exercise}

\begin{exercise}
Let $a, b, n \in \N$.  Prove that $a|b$ if and only if $a^n | b^n$.  [{\em Suggestion:}  For the reverse implication, write $a=(a,b)r$ and $b=(a,b)s$ with $(r,s)=1$.  Apply the preceding exercise and conclude that $r=1$.]
\end{exercise}

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