Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.3 - Page 178: 21

Answer

Let $a$ and $b$ be two even integers. Then by the definition of even, $a=2m$ and $b=2n$ for some integers $m$ and $n$. Therefore, $ab=(2m)(2n)=4mn$. But $4mn$ is divisible by $4$, since $mn$ is a product of integers and therefore an integer. Therefore, $ab$ is divisible by $4$ by the definition of divisibility. Since $a$ and $b$ were arbitrarily chosen, it must be that the product of any two even integers is divisible by $4$.

Work Step by Step

Note that "reusing" variables (e.g., letting $a=2n$ and $b=2n$) is prohibited, because then the proof would only be valid when $a=b$. For more on this, see section 4.1.
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