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\centerline{\textbf{Deceptively Uninspiring Homework 2}}
\centerline{Due Wednesday April 12th at the beginning of class}
\bigskip
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 Prove each of the following. For this exercise only, write your proofs in table form (like in Example~1.2 in the Conroy-Taggart) with a column for each {\bf Step} and its {\bf Justification}. Your Justifications may be any of the Axioms of the Integers or a previous part of this exercise.
\begin{parts}
\part If $a$ and $b$ are integers, then $(-a)\cdot b=-(ab)$.\\
%Answer goes here.
\part If $a$ and $b$ are integers, then $(a+b)^2 = a^2 +2ab + b^2.$\\
%Answer goes here.
\part If $a+b=a$, then $b=0$.\\
%Answer goes here.
\part If $a$ is an integer, then $a\cdot 0=0$.\\
%Answer goes here.
\end{parts}
\question Suppose $a$ and $b$ are integers. Prove each of the following.\\
\begin{parts}
\part If $a$ is even and $b$ is odd, then $a+b$ is odd.\\
%Answer goes here.
\part If $a$ and $b$ are both odd, then $a+b$ is even.\\
%Answer goes here.
\part If $a+b$ is odd, then $a$ and $b$ have opposite parity.\\
%Answer goes here.
\end{parts}
\question Suppose $a$ and $b$ are {\bf negative} integers. Prove that, if $a**b^2.$\\
%Answer goes here.
\question Suppose $a$ and $b$ are {\bf positive} integers. Prove that, if $a \mid b$, then $a\leq b$. \\
%Answer goes here.
\question Suppose $a>0$ and $b\geq 0$ are integers such that $a \mid b$. Prove that, if $b**