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Show that the following statement is false: For all integers a and b, if 3 (a + b) then 3|(a - b).

Show that the following statement is false: For all integers a and b, if 3 (a + b) then 3|(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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Certainly! Below is a transcription of the text suitable for an educational website: --- **Chapter 4: Elementary Number Theory and Methods** Page 198 18. Consider the following statement: The negative of any multiple of 3 is a multiple of 3. a. Write the statement formally using a quantifier and a variable. b. Determine whether the statement is true or false and justify your answer. 19. Show that the following statement is false: For all integers \(a\) and \(b\), if \(3 | (a + b)\) then \(3 | (a - b)\). For each statement in 20–32, determine whether the statement is true or false. Prove the statement directly from the definitions if it is true, and give a counterexample if it is false. H 20. The sum of any three consecutive integers is divisible by 3. 21. The product of any two even integers is a multiple of 4. --- This transcription is used to support educational endeavors by providing formatted mathematical statements and exercises for students to explore number theory concepts further.
Transcribed Image Text:Certainly! Below is a transcription of the text suitable for an educational website: --- **Chapter 4: Elementary Number Theory and Methods** Page 198 18. Consider the following statement: The negative of any multiple of 3 is a multiple of 3. a. Write the statement formally using a quantifier and a variable. b. Determine whether the statement is true or false and justify your answer. 19. Show that the following statement is false: For all integers \(a\) and \(b\), if \(3 | (a + b)\) then \(3 | (a - b)\). For each statement in 20–32, determine whether the statement is true or false. Prove the statement directly from the definitions if it is true, and give a counterexample if it is false. H 20. The sum of any three consecutive integers is divisible by 3. 21. The product of any two even integers is a multiple of 4. --- This transcription is used to support educational endeavors by providing formatted mathematical statements and exercises for students to explore number theory concepts further.
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19. To show that the statement is false, it is sufficient to find integers a and b such that 3 divides a+b, but 3 does not divide a-b.

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