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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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12. If n = 4k+ 1, does 8 divide n² - 1? 13. If n = 4k+ 3, does 8 divide n² - 1? Copyright 2020 Cengage Learning. All Rights Reserved. May not be cop 198 CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF 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 counterex- ample if it is false. H 20. The sum of any three consecutive integers is divis- ible by 3. 21. The product of any two even integers is a multiple of 4. 34. 35. 1 36. I i 1 E i ii
Transcribed Image Text:12. If n = 4k+ 1, does 8 divide n² - 1? 13. If n = 4k+ 3, does 8 divide n² - 1? Copyright 2020 Cengage Learning. All Rights Reserved. May not be cop 198 CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF 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 counterex- ample if it is false. H 20. The sum of any three consecutive integers is divis- ible by 3. 21. The product of any two even integers is a multiple of 4. 34. 35. 1 36. I i 1 E i ii
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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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