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\noindent{\Large{\textsc{Number Theory\hfill Assignment 5.1  \\ Spring 2022 \hfill Due March 23}}}

\large

\vspace{1cm}

\begin{exercise}
Let $n \in \N$ and write $n=ab^2$ with $a,b \in \N$ and $a$ squarefree.  Prove that $a$ and $b$ are unique.  That is, show that if $a,b,c,d \in \N$, $a$ and $c$ are squarefee, and $ab^2 = cd^2$, then $a=c$ and $b=d$.
\end{exercise}

\begin{exercise}
Let $k \in \N$, $k \ge 2$.  We say that $n \in \N$ is {\em $k$th power free} if for every $d \in \N$, $d^k | n$ implies $d=1$.  Prove that every $n \in \N$ can be expressed uniquely in the form $n = a b^k$, where $a, b \in \N$ and $a$ is $k$th power free. 
\end{exercise}

\begin{exercise}
Let $k \ge 2$ be an integer.  Prove that $n \in \N$ is $k$th power free if and only if $\nu_p(n) < k$ for all primes $p$.
\end{exercise}

\begin{exercise}
If $\mu: \N \to \{ 0, \pm 1\}$ denotes the M\"{o}bius function, prove that for every $n \in \N$ one has
\[
\sum_{d^k | n} \mu(d) = \begin{cases} 1 & \text{ if } n \text{ is } $k$ \text{th power free,}\\ 0 & \text{ otherwise.} \end{cases}
\]
Here the sum is indexed by all $d \in \N$ for which $d^k | n$.
\end{exercise}

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