Number 18 31. Suppose the division algorithm applied to a and b yields a = qb +r. Proveabout how a direct proof would work. In most cases contrapositive is easier.)A. Prove the following statements with contrapositive proof. (In each case, thinkContraposttive Proof136Exercises for Chapter 51. Suppose neZ. If n is even, then n is even.2. Suppose n eZ. If n2 is odd, then n is odd.S. Suppose a, be Z. If a2(b2 -2b) is odd, then a and b are odd4. Suppose a, b,c € Z. If a does not divide be, then a does not divid,.5. Suppose x ER. If x2 + 5x -1.7. Suppose a,beZ. If both ab and a +b are even, then both a and8. Suppose x ER. If x - 4x* +3x3-x2 + 3x -420, then x 20.9. Suppose n e Z. If 3 n2, then 3łn.10. Suppose x,y,z € Z and x#0. If xyz, then xy and x z.11. Suppose x,y€ Z. If x2(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 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 xe 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. Ifn is odd, then 81 (n2- 1).18) If a,beZ, 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 ne N. 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 a2 = ab (mod n).(24, If a = b (mod n) and c = 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 ode27. 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-6,01.30. If a = b (mod n), then ged(a,n)= gcd(b,n).%3D%3Dgod(a b)

Question

Number 18 31. Suppose the division algorithm applied to a and b yields a = qb +r. Proveabout how a direct proof would work. In most cases contrapositive is easier.)A. Prove the following statements with contrapositive proof. (In each case, thinkContraposttive Proof136Exercises for Chapter 51. Suppose neZ. If n is even, then n is even.2. Suppose n eZ. If n2 is odd, then n is odd.S. Suppose a, be Z. If a2(b2 -2b) is odd, then a and b are odd4. Suppose a, b,c € Z. If a does not divide be, then a does not divid,.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 and8. Suppose x ER. If x - 4x* +3x3-x2 + 3x -420, then x 20.9. Suppose n e Z. If 3 n2, then 3łn.10. Suppose x,y,z € Z and x#0. If xyz, then xy and x z.11. Suppose x,y€ Z. If x2(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 > 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 xe 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. Ifn is odd, then 81 (n2- 1).18) If a,beZ, 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 ne N. 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 a2 = ab (mod n).(24, If a = b (mod n) and c = 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 ode27. 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-6,01.30. If a = b (mod n), then ged(a,n)= gcd(b,n).%3D%3Dgod(a b)

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(18) If a, be Z, then (a + b)3 = a3 + b3 (mod 3).