To prove the statement " 8 does not divide x² – 1 if x is even." directly. (choose the correct answer of the proof) Assume that Ox is odd. Ox is even. 08 does not divide x2 – 1 and x is odd. 08 divides x² – 1 and x is even. 08 divides x² – 1. then Ox = 2k + 1. Ox = 2k for some integer k. Ox² – 1 = 8k. 08 = k(x² – 1) So Ox² – 1 = 4k² + 4k = 4k(k + 1). Since k(k + 1) is even then 8 divides x2 – 1. Hence, result follows. Ox² – 1 = 4k² –1 = 4k? – 2+1 = 2(2k² – 1) + 1 which is odd. So 8 does not divide x2 – 1. Hence, result follows. - Ox² = 8k + 1 = 2(4k) + 1. So x² is odd and so is x. Hence, result follows. 08 = k(x – 1)(x + 1) sincex – 1 and x + 1 have the same parity then if they are both odd and k is odd we get a contradiction. So result follows.

To prove the statement " 8 does not divide x² – 1 if x is even." directly. (choose the correct answer of the proof) Assume that Ox is odd. Ox is even. 08 does not divide x2 – 1 and x is odd. 08 divides x² – 1 and x is even. 08 divides x² – 1. then Ox = 2k + 1. Ox = 2k for some integer k. Ox² – 1 = 8k. 08 = k(x² – 1) So Ox² – 1 = 4k² + 4k = 4k(k + 1). Since k(k + 1) is even then 8 divides x2 – 1. Hence, result follows. Ox² – 1 = 4k² –1 = 4k? – 2+1 = 2(2k² – 1) + 1 which is odd. So 8 does not divide x2 – 1. Hence, result follows. - Ox² = 8k + 1 = 2(4k) + 1. So x² is odd and so is x. Hence, result follows. 08 = k(x – 1)(x + 1) sincex – 1 and x + 1 have the same parity then if they are both odd and k is odd we get a contradiction. So result follows.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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