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8. Suppose a, b e Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b (mod 5).

8. Suppose a, b e Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b (mod 5).

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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Detailed prove for number 8
ch.iastate.edu/class/BookOfProof.pdf Exercises for Chapter 7 Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters 4-7. 1. Suppose x e Z. Then x is even if and only if 3x+5 is odd. 2. Suppose xe Z. Then x is odd if and only if 3x + 6 is odd. 3. Given an integer a, then a³ + a² + a is even if and only if a is even. 4. Given an integer a, then a² + 4a +5 is odd if and only if a is even. 5. An integer a is odd if and only if a³ is odd. 6. Suppose x,y e R. Then x³ + x²y = y² + xy if and only if y=x² or y=-x. Then (x + y)² = x² + y² if and only if x = 0 or y = 0. 7. Suppose x,y e R. 8. Suppose a,b € Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b (mod 5). 9. Suppose a € Z. Prove that 14 | a if and only if 7|a and 2 a. 10. If a € Z, then a³ = a (mod 3). F4 11. Suppose a, b e Z. Prove that (a-3)62 is even if and only if a is odd or b is even. 12. There exist a positive real number x for which x² <√√x. 13. Suppose a, b e Z. If a + b is odd, then a² + b² is odd. 14. Suppose a € Z. Then a² la if and only if a € (-1,0,1}. 15. Suppose a, b € Z. Prove that a + b is even if and only if a and b have the same parity. 16. Suppose a, b e Z. If ab is odd, then a² + b² is even. 17. There is a prime number between 90 and 100. 18. There is a set X for which NE X and N≤X. 19. If ne N, then 20 +2¹+22 +23+24+...+2 = 2n+1 -1. 20. There exists an neN for which 11 (2-1). !! A -O F6 CD A -.. F7 HI ų F8 (T) F9 prt sc F10 home 8
Transcribed Image Text:ch.iastate.edu/class/BookOfProof.pdf Exercises for Chapter 7 Prove the following statements. These exercises are cumulative, covering all techniques addressed in Chapters 4-7. 1. Suppose x e Z. Then x is even if and only if 3x+5 is odd. 2. Suppose xe Z. Then x is odd if and only if 3x + 6 is odd. 3. Given an integer a, then a³ + a² + a is even if and only if a is even. 4. Given an integer a, then a² + 4a +5 is odd if and only if a is even. 5. An integer a is odd if and only if a³ is odd. 6. Suppose x,y e R. Then x³ + x²y = y² + xy if and only if y=x² or y=-x. Then (x + y)² = x² + y² if and only if x = 0 or y = 0. 7. Suppose x,y e R. 8. Suppose a,b € Z. Prove that a = b (mod 10) if and only if a = b (mod 2) and a = b (mod 5). 9. Suppose a € Z. Prove that 14 | a if and only if 7|a and 2 a. 10. If a € Z, then a³ = a (mod 3). F4 11. Suppose a, b e Z. Prove that (a-3)62 is even if and only if a is odd or b is even. 12. There exist a positive real number x for which x² <√√x. 13. Suppose a, b e Z. If a + b is odd, then a² + b² is odd. 14. Suppose a € Z. Then a² la if and only if a € (-1,0,1}. 15. Suppose a, b € Z. Prove that a + b is even if and only if a and b have the same parity. 16. Suppose a, b e Z. If ab is odd, then a² + b² is even. 17. There is a prime number between 90 and 100. 18. There is a set X for which NE X and N≤X. 19. If ne N, then 20 +2¹+22 +23+24+...+2 = 2n+1 -1. 20. There exists an neN for which 11 (2-1). !! A -O F6 CD A -.. F7 HI ų F8 (T) F9 prt sc F10 home 8
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