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 4.6 - Page 205: 4


We prove by contradiction. Suppose that $7|(7m+4)$ for some integer $m$. Then by the definition of "divides," $7m+4=7k$ for some integer $k$. Subtracting $7k$ and $4$ from both sides, we get $7m-7k=-4$. Dividing both sides by $7$, we get $m-k=-\frac{4}{7}$. But this is a contradiction, because the integers are closed under subtraction, yet $-\frac{4}{7}$ is not an integer. Hence, our assumption must be false, and we conclude that, for all integers $m$, $7m+4$ is not divisible by $7$.

Work Step by Step

Recall that the closure properties of the integers state that the sum, difference, and product of integers is always an integer.
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