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\centerline{\textbf{Deceptively Uninspiring Homework 3}}
\centerline{Due Wednesday April 19th at the beginning of class}
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You may handwrite or type your answers/solutions/proofs. I highly encourage the use of a mathematical typesetting language (like \LaTeX). If you handwrite, please make sure that your work is legible, and please staple your homework when you turn them in.
\begin{questions}
\question Suppose $a$ and $b$ are integers. Prove that $ab+a+b$ is even if and only if both $a$ and $b$ are even.
\question Suppose $n$ is an integer. Prove that $n^2$ is even if and only if $4\mid n^2$.
\question Show that $6\mid n(n+1)(2n+1)$ for every positive integer $n$.
\question Show that $\sqrt{3}$ is irrational.
\question If $x$ is irrational, show that $x+\frac{a}{b}$ is irrational for all $a,b\in \mathbb{Z}\backslash \{0\}$.
\question If $x$ is irrational, show that $x\cdot \frac{a}{b}$ is irrational for all $a,b\in \mathbb{Z}\backslash \{0\}$.
\end{questions}
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