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\begin{document}
\title{Math 301 Homework}
\author{Mary Radcliffe}
\date{Due 1 February 2018}
\maketitle

Complete the following problems. Fully justify each response.

\begin{enumerate}

\item Let $n$ be a natural number. Prove that $n$ is divisible by 3 if and only if the sum of the base-10 digits of $n$ is divisible by 3.

\item Let $x\in \R$. Prove that if $x\neq 0$, then $x^2>0$.

\item Let $x$ be an irrational number. Prove that $\frac{1}{x}$ and $-x$ are both irrational.

\item Prove that for any real number $x\in \R$, there exists a real number $y\in \R$ such that $x+y\in \Z$.


\item Let $p(n)$ be the statement ``$2n+1$ is divisible by $3$,'' for $n\in \N$. Prove that for any $n\in \N$, $p(3n+1)$ is true, and $p(3n)$ and $p(3n+2)$ are false. (Note: this is Exercise 1.3.9 on page 50. There is some further discussion there about the statement $p(n)$.)

\item Explain, in your own words, the Principle of Weak Induction (Thm. 1.3.10). Without writing a formal proof, discuss why the theorem is true.

\end{enumerate}
\end{document}