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{\bf CSci 243 Homework 3}
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**My name**  %update me!!
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\begin{enumerate}

\item (10 points)  Prove that if $n$ is an integer and $n^3 + 5$ is odd, then $n$ is odd using
\begin{enumerate}
\item a proof by contraposition.
\item a proof by contradiction.
\end{enumerate}

\item (10 points) Prove that $n^2$ can be divided by 3 if and only if $n$ can be divided by 3.

\item (6 points) Disprove the statement that ``Every positive integer can be written as the sum of the squares of three integers'' by finding a counterexample.

\item (6 points) Use a proof by cases to show that $min(a,min(b,c))=min(min(a,b),c)$ whenever $a, b, c$ are real numbers.

\item (6 points) Prove using the notion of without loss of generaility that $5x+5y$ is an odd integer when $x$ and $y$ are integers of opposite parity.


\item (6 points) A prime number is a positive integer greater than 1 that is only divisible by 1 and itself.
Prove or disprove that $\sqrt[k]p$ is irrational for any prime number $p$ and any integer $k\ge2$.  
Hint: Euclid's Lemma- If a prime divides the \textit{product} of two integers, it must divide at least one of those integers.

\item (6 points) Suppose that $a$ and $b$ are odd integers with $a \ne b$. Show there is a unique integer $c$ such that $|a - c| = |a - b|$.

\item (\textbf{Bonus Question} 1 points\footnote{The points for bonus questions are added directly to the final grade.}) Show that $\sqrt3$ is irrational. 

\end{enumerate}

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