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23. Let a,be Z and n eN. If a =b (mod n), then a = ab (mod n). (24, If a b (mod n) and c=d (mod n), then ac = bd (mod n). 25. Let neN. If 2"-1 is prime, then n is prime.

23. Let a,be Z and n eN. If a =b (mod n), then a = ab (mod n). (24, If a b (mod n) and c=d (mod n), then ac = bd (mod n). 25. Let neN. If 2"-1 is prime, then n is prime.

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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Question
Number 24
31. Suppose the division algorithm applied to a and b yields a = qb + r. Prove about how a direct proof would work. In most cases contrapositive is easier.) A. Prove the following statements with contrapositive proof. (In each case, think Contrapositive Proof 136 Exercises for Chapter 5 1. Suppose nEZ. If n is even, then n is even. 2. Suppose n €Z. Ifn2 is odd, then n is odd. 3. Suppose a, be Z. If a2(b2 -2b) is odd, then a and b are odd 4. Suppose a, b,e € Z. If a does not divide be, then a does not divide 5. Suppose x ER. If x2 + 5x<0 then x<0. 6. Suppose x ER. If x -x> 0 then x>-1. 7. Suppose a,beZ. If both ab and a + b are even, then both a and i 8. Suppose x ER. If x -4x*+3x-x2 +3x- 4>0, then x >0. 9. Suppose n e Z. If 3 n2, then 3 n. 10. Suppose x,y,z eZ and x#0. If xyz, then xty and xtz. 11. Suppose x,ye Z. If x(y+3) is even, then x is even or y is odd. 12. Suppose a € Z. If a? is not divisible by 4, then a is odd. 13. Suppose xER. If r+7x³ + 5x > x* + x2 + 8, then x 2 0. B. Prove the following statements using either direct or contrapositive proof. 14. If a,beZ and a and b have the same parity, then 3a +7 and 76 -4 do not. 15. Suppose x€ Z. If x-1 is even, then x is odd. 16. Suppose x, y e Z. If x+ y is even, then x and y have the same parity. 17. If n is odd, then 81 (n2-1). 18) If a,be Z, then (a+b) = a³ +b3 (mod 3). 19. Let a,b, CEZ and n e N. If a = b (mod n) and a = c (mod n), then c = b (mod n). 20. If a eZ and a = 1 (mod 5), then a? = 1 (mod 5). 21. Let a,beZ and neN. If a =b (mod n), then a = b (mod n). 22. Let a e Z, neN. If a has remainder r when divided by n, then a =r (mod n). 23. Let a,be Z and neN. If a =b (mod n), then a = ab (mod n). (24, If a = b (mod n) and e = d (mod n), then ac = bd (mod n). 25. Let n e N. If 2"-1 is prime, then n is prime. 26. If n 24-1 for keN, then every entry in Row n of Pascal's Triangle 1s ode 27. If a = 0 (mod 4) or a = 1 (mod 4), then () is even. 28. If ne Z, then 4 (n2-3). 29. If integers a and b are not both zero, then ged(a, b) = gcd(a-0,01. 30. If a = b (mod n), then ged(a,n)=gcd(b,n). %3D godla b)
Transcribed Image Text:31. Suppose the division algorithm applied to a and b yields a = qb + r. Prove about how a direct proof would work. In most cases contrapositive is easier.) A. Prove the following statements with contrapositive proof. (In each case, think Contrapositive Proof 136 Exercises for Chapter 5 1. Suppose nEZ. If n is even, then n is even. 2. Suppose n €Z. Ifn2 is odd, then n is odd. 3. Suppose a, be Z. If a2(b2 -2b) is odd, then a and b are odd 4. Suppose a, b,e € Z. If a does not divide be, then a does not divide 5. Suppose x ER. If x2 + 5x<0 then x<0. 6. Suppose x ER. If x -x> 0 then x>-1. 7. Suppose a,beZ. If both ab and a + b are even, then both a and i 8. Suppose x ER. If x -4x*+3x-x2 +3x- 4>0, then x >0. 9. Suppose n e Z. If 3 n2, then 3 n. 10. Suppose x,y,z eZ and x#0. If xyz, then xty and xtz. 11. Suppose x,ye Z. If x(y+3) is even, then x is even or y is odd. 12. Suppose a € Z. If a? is not divisible by 4, then a is odd. 13. Suppose xER. If r+7x³ + 5x > x* + x2 + 8, then x 2 0. B. Prove the following statements using either direct or contrapositive proof. 14. If a,beZ and a and b have the same parity, then 3a +7 and 76 -4 do not. 15. Suppose x€ Z. If x-1 is even, then x is odd. 16. Suppose x, y e Z. If x+ y is even, then x and y have the same parity. 17. If n is odd, then 81 (n2-1). 18) If a,be Z, then (a+b) = a³ +b3 (mod 3). 19. Let a,b, CEZ and n e N. If a = b (mod n) and a = c (mod n), then c = b (mod n). 20. If a eZ and a = 1 (mod 5), then a? = 1 (mod 5). 21. Let a,beZ and neN. If a =b (mod n), then a = b (mod n). 22. Let a e Z, neN. If a has remainder r when divided by n, then a =r (mod n). 23. Let a,be Z and neN. If a =b (mod n), then a = ab (mod n). (24, If a = b (mod n) and e = d (mod n), then ac = bd (mod n). 25. Let n e N. If 2"-1 is prime, then n is prime. 26. If n 24-1 for keN, then every entry in Row n of Pascal's Triangle 1s ode 27. If a = 0 (mod 4) or a = 1 (mod 4), then () is even. 28. If ne Z, then 4 (n2-3). 29. If integers a and b are not both zero, then ged(a, b) = gcd(a-0,01. 30. If a = b (mod n), then ged(a,n)=gcd(b,n). %3D godla b)
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