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1. For each of the following statements: if the statement is true, then give a proof; if the statement is false, then write out the negation and prove that. (a) There exists an integer n, so that n³ - n is odd. (b) √6 is irrational. (c) For all a, b = Z, if a > 1 and b > 1, then gcd (2a, 2b) = 2 gcd(a, b).

1. For each of the following statements: if the statement is true, then give a proof; if the statement is false, then write out the negation and prove that. (a) There exists an integer n, so that n³ - n is odd. (b) √6 is irrational. (c) For all a, b = Z, if a > 1 and b > 1, then gcd (2a, 2b) = 2 gcd(a, b).

Advanced Engineering Mathematics
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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1. For each of the following statements: if the statement is true, then give a proof; if the statement is false, then write out the negation and prove that. (a) There exists an integer n, so that n³ - n is odd. (b) √6 is irrational. (c) For all a, b = Z, if a > 1 and b > 1, then gcd (2a, 2b) = 2 gcd(a, b).
Transcribed Image Text:1. For each of the following statements: if the statement is true, then give a proof; if the statement is false, then write out the negation and prove that. (a) There exists an integer n, so that n³ - n is odd. (b) √6 is irrational. (c) For all a, b = Z, if a > 1 and b > 1, then gcd (2a, 2b) = 2 gcd(a, b).
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Let Z+ be the set of all positive integers. (a) Use the Euclidean Algorithm to compute gcd(2023, 271) and use that to find integers x and y so that ged(2023, 271) = 2023 x + 271 y. (b) Find integers m and n so that ged(2023, 271) = 2023 m +271 n, but m‡ x and n ‡ y. Note that z and y are the integers that you found in part (a). (c) Is it true that: for all a, b € Z+, if a >1 and b> 1, then gcd(a, b) < < ged(a³,6³)? Prove your answer.
Transcribed Image Text:Let Z+ be the set of all positive integers. (a) Use the Euclidean Algorithm to compute gcd(2023, 271) and use that to find integers x and y so that ged(2023, 271) = 2023 x + 271 y. (b) Find integers m and n so that ged(2023, 271) = 2023 m +271 n, but m‡ x and n ‡ y. Note that z and y are the integers that you found in part (a). (c) Is it true that: for all a, b € Z+, if a >1 and b> 1, then gcd(a, b) < < ged(a³,6³)? Prove your answer.
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