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: 22

Answer

Let $n$ be an arbitrary integer. By the definition of a necessary condition, $n$ being divisible by $2$ is a necessary condition for $n$ being divisible by $6$ means that [$n$ is divisible by $6$] implies [$n$ is divisible by $2$]. Assume that $n$ is divisible by $6$. Then $n=6k=2(3k)$ for some integer $k$. But $3k$ is also an integer, and since $n=2(3k)$, it must be that $n$ is divisible by $2$. Since $n$ was an arbitrarily chosen integer divisible by $6$, it must be that every integer divisible by $6$ is divisible by $2$, i.e., divisibility by $2$ is a necessary condition for divisibility by $6$.

Work Step by Step

Recall from the chapters on logic that "B is a necessary condition for A" and "A is a sufficient condition for B" both mean the same thing as "A implies B."
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