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 4.1 - Page 162: 41


The objective is to prove that the product of m and n is even, however the proof starts out by supposing it is even when it should be proved first. Therefore, the use of "Thus, mn = (2p)(2q + 1) = 2r" in the conclusion is completely unjustified.

Work Step by Step

Let m be an even integer, by definition of even integers, we write m as → m=2p now, let n be an odd integer, by definition of odd integers, we write n as → n=2q+1 consider their product to be, mn = (2p)(2q+1) = 4pq + 2p = 2(pq+p) as the product of two integers (pq) is odd and by adding an integer (pq) to another integer (p), we obtain another integer (pq+p), so we can say that (pq+p) is also an integer. let the integer 2(pq+p) = r, therefore, mn = 2r, as 2r is an even integer by the definition of even integers, the product of mn is even!
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