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\ihead{Math 221 Winter 2025 (Darij Grinberg): homework set 5}
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\begin{document}

\begin{center}
\textbf{Math 221 Winter 2025 (Darij Grinberg): homework set 5}

due date: Thursday 2025-02-20 at 11:59PM on gradescope (
\url{https://www.gradescope.com/courses/930212} ).

Please solve only \textbf{3 of the 6 exercises}.
\end{center}



\begin{exercise}
\label{exe.ent.gcd-monoton}Let $a_{1},a_{2},b_{1},b_{2}\in\mathbb{Z}$ be
integers satisfying $a_{1}\mid b_{1}$ and $a_{2}\mid b_{2}$. Prove that
$\gcd\left(  a_{1},a_{2}\right)  \mid\gcd\left(  b_{1},b_{2}\right)  $.
\end{exercise}



\begin{exercise}
\label{exe.ent.gcd(ak,bk)}Let $a,b\in\mathbb{Z}$ and $k\in\mathbb{N}$. Prove
that%
\[
\gcd\left(  a^{k},b^{k}\right)  =\left(  \gcd\left(  a,b\right)  \right)
^{k}.
\]

\end{exercise}

\begin{exercise}
\label{exe.ent.a!b!-div-a+b-1!}Let $a$ and $b$ be two coprime positive
integers. Prove that%
\[
a!\cdot b!\mid\left(  a+b-1\right)  !.
\]


[\textbf{Hint:} Let $g=\dfrac{\left(  a+b-1\right)  !}{a!\cdot b!}$. Prove
that both $ag$ and $bg$ are integers. What then?]
\end{exercise}

\bigskip

The following exercise is a generalization of the prime divisor separation
theorem (Theorem 3.6.5):

\begin{exercise}
\label{exe.24hw4.1}Let $p$ be a prime, and let $m\in\mathbb{N}$. Let $a$ and
$b$ be two integers such that $p^{m}\mid ab$ and $p^{m}\nmid a$. Prove that
$p\mid b$.
\end{exercise}



Next comes a generalization of the $p\mid\dbinom{p}{k}$ divisibility (Theorem 3.6.3):

\begin{exercise}
\label{exe.23hw4.6}Let $p$ be a prime. Let $m\in\mathbb{N}$, and let
$k\in\left\{  1,2,\ldots,p^{m}-1\right\}  $. Prove that $p\mid\dbinom{p^{m}%
}{k}$.
\end{exercise}



\begin{exercise}
Let $p$ and $q$ be two distinct primes. Prove that $p^{q}+q^{p}\equiv
p+q\operatorname{mod}pq$ and $p^{q-1}+q^{p-1}\equiv1\operatorname{mod}pq$.

[\textbf{Hint:} First show that $p$ and $q$ are coprime.]
\end{exercise}


\end{document}