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\parbox{2.25in}{\bf Math 2110}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Oct. $7^\mathrm{th}$, 2020}

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%\noindent {\large\bf Name:} \underline{\Large\quad My Name Goes Here \quad} \hfill 

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\begin{description}
\item[\#1] Let $n$  be a natural number and suppose $n^2$ is odd. Prove that $n$ is odd.

\item[\#2] Prove that there is no largest natural number.

\item[\#3] Prove that $\log_{10}(2)$ is irrational. \quad {\it Hint:} The Fundamental Theorem of Arithmetic (integers greater than one have {\it unique} prime factorizations) should help at some point in your proof.

\item[\#4] Prove Theorem 20 (page 80): Let $p$ be a natural number bigger than $1$. If there is no natural number $m$ dividing $p$ such that $1<m \leq \sqrt{p}$, then $p$ is prime.

%\item[\#5] Let $a,b$ denote the lengths of the legs of a non-degenerate right triangle (non-degenerate means $a,b \not=0$) and let $c$ denote the length of the hypotenuse of this triangle. Prove that $a+b > c$.

\item[\#5] Suppose that $r \in \mathbb{R}$ (i.e., $r$ is a real number). It is the case that there is some $n \in \mathbb{Z}$ (i.e., integer $n$) such that $n \leq r < n+1$ (you don't have to prove this).\footnote{{\bf Existence Proof:} Let $n=\lfloor r \rfloor$ where $\lfloor r \rfloor$ is the floor function (i.e., the largest integer less than or equal to $r$). When $r \geq 0$, $\lfloor r \rfloor$ is just the integer part of $r$ (i.e., $\lfloor n.d_1d_2d_3\cdots \rfloor = n$) and if $r<0$, then $\lfloor r \rfloor$ is the integer part minus $1$ (e.g., $\lfloor -3.134 \rfloor = -3-1=-4$). In other words, the floor function just rounds down to the nearest integer. Thus by definition $\lfloor r \rfloor$ is an integer and $\lfloor r \rfloor \leq r$. Notice that if $\lfloor r \rfloor+1 \leq r$, then $\lfloor r \rfloor$ is not the largest integer less than or equal to $r$ (contradiction). Therefore, $r<\lfloor r \rfloor+1$ and so $n=\lfloor r \rfloor$ is an integer such that $n \leq r < n+1$. $\blacksquare$} Prove that such an integer is unique (i.e., there is only one integer $n$ such that $n \leq r < n+1$).

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