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\centerline{\rm \bf  Math 301   Homework 2 (due on September 6, 2023)}\vskip.1in

\noindent{\Large\bf Section 1.2}\vskip.1in

\noindent{\bf 1.} 
Find the greatest common divisor using Euclid's algorithm.
\begin{itemize}
\item[d)] $(143, 231)$
\item[h)] $(12378, 3054)$
\end{itemize}
\vskip.1in

\noindent{\bf 4.}
\begin{itemize}
\item[a)] If $a|b$ and $a|c$, prove that $a|(b+c)$.
\item[b)] If $a|b$ and $a|c$, prove that $a|(br+ct)$ for any $r,t\in\Z$.
\end{itemize}
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\noindent{\bf 22.} 
If $(a, c) = 1$ and $(b, c) = 1$, prove that $(ab, c) = 1$.
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\noindent{\bf 24.} 
Let $a, b, c \in \Z$. Prove that the equation $ax+by=c$ has integer solutions if and only if $(a, b)| c$.
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\vskip.2in
\noindent{\Large\bf Section 1.3}
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\noindent{\bf 14.} 
Let $p$ be an integer other than $0,\pm 1$ with this property: Whenever $b$ and $c$ are integers such that $p|bc$, then $p|b$ or $p|c$. Prove that $p$ is prime.\\
\emph{Hint:} If $d$ is a divisor of $p$, say $p=dt$, then $p|d$ or $p|t$. Show that this implies $d=\pm p$ or $d=\pm 1$.
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\noindent{\bf 22.} 
Let $n=p_1^{r_1}p_2^{r_2}\cdots p_n^{r_n}$ where $p_i$ are distinct primes and each $r_i>0$. Prove that $n$ is a perfect square if and only if each $r_i$ is even.
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\noindent{\bf 30.} 
\begin{itemize}
\item[a)] Prove that there are no nonzero integers $a,b$ such that $a^2=2b^2$.\\
\emph{Hint: Use the Fundamental Theorem of Arithmetic.}
\item[b)] Prove that $\sqrt{2}$ is irrational.\\
\emph{Hint: Use proof by contradiction (Appendix A). Assume that $\sqrt{2}=a/b$ (with $a,b\in\Z$) and use part (a) to reach a contradiction.}
\end{itemize}
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\noindent{\bf 34.} 
Prove or disprove: If $n$ is an integer and $n>2$, then there exists a prime $p$ such that $n<p<n!$.
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\noindent{\bf 36.} 
Let $p, q$ be primes with $p\ge 5$, $q\ge 5$. Prove that $24|(p^2-q^2)$.
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