Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 1 - Section 1.7 - Introduction to Proofs - Exercises - Page 91: 16

Answer

See direct proof below.

Work Step by Step

Let $m, n\in \mathbb{Z}$ and $mn$ be even. Then $mn$ has a factor of 2. $mn=2\frac{mn}{2}$. By factoring, either $\frac{m}{2}$ or $\frac{n}{2}$ is an integer. Without loss of generality, let this be $\frac{m}{2}$. Because $m=2*\frac{m}{2}$, $\frac{m}{2}\in \mathbb{Z}$, $m$ has a factor of 2 and thus is even.
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