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\begin{center}
{\Huge
15-151 Homework 7
}

Please submit in class at 8:00am on Thursday 27th July
\end{center}

\subsection*{Exercises}

\begin{enumerate}[1.]
\item Let $p$ and $q$ be distinct prime integers and let $k,\ell,m,n \in \mathbb{N}$. Prove that

$ $ \hfill $\mathrm{gcd}(p^kq^{\ell}, p^mq^n) = p^{\mathrm{min}(k,m)}q^{\mathrm{min}(\ell,n)}$
\points{7}

\item Let $a,p \in \mathbb{Z}$ and suppose that $p$ is prime. Prove that $a \perp p$ if and only if $p \nmid a$.
\points{7}

\item Let $a,b \in \mathbb{Z}$ and let $d=\mathrm{gcd}(a,b)$. Prove that $\frac{a}{d} \perp \frac{b}{d}$.
\points{7}

\item Let $n \in \mathbb{Z}$ with $n > 2$. Prove that the set
$$\{ k \in \mathbb{Z} \mid n < k < n! \}$$
contains a prime number, where $n! = 1 \times 2 \times \cdots \times n$ is the \textit{factorial} of $n$. 
\points{7}

\textit{Hint: consider the prime factors of $n!-1$.}

\item Let $n \in \mathbb{Z}$ with $n>2$. Prove that $n=m^2$ for some $m \in \mathbb{Z}$ if and only if the multiplicity of each prime in the prime factorisation of $n$ is even.
\points{7}
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