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\title{CSCI 301 - Assignment 3, Fall 2024}
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your solutions. \\
This is an individual assignment. See the syllabus for the collaboration policy.\\

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In the Lecture 10 Exercises, we proved the following two propositions, which we number here for ease of reference:

\textbf{Proposition 1:} If two integers have the same parity, then their sum is even.

\textbf{Proposition 2:} If two integers have opposite parity, then their sum is odd.

In Problems 1 and 2, you will prove the corresponding theorems for the \textit{product} of two integers. In Problem 3, ou are then free to make use of these four propositions in support of your proof.

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Prove each of the following propositions. Consider using direct proof, contrapositive proof, or proof by contradiction; try to use the approach that makes for the simplest proof.
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\item (5 points) \textbf{Propostion 3:} If $a$ and $b$ have the same parity, then $ab$ also has the same parity.

\textbf{Proof:} Replace this text with your proof.

\item (5 points) \textbf{Propostion 4:} If $a$ and $b$ have opposite parity, then $ab$ is even.

\textbf{Proof:} Replace this text with your proof.

\item (10 points) Suppose $a$ and $b$ are integers. Then $a^2(b+3)$ is even if and only if $a$ is even or $b$ is odd. Follow the outline for If-and-Only-If Proof from the Book of Proof Chapter 7.1, page 147. \textit{Hint}: Given that this proof consists of two parts, you may find it helpful to consider different proof strategies for each part.

\textbf{Proof:} Replace this text with your proof.

\item \textbf{Bonus Proposition} (extra credit, up to 2 points): In any right triangle with integer side lengths, at least one of the non-hypotenuse edges has even length.
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