B. Prove the following statements using either direct or contrapositive proof. 14. If a, b e Z and a and b have the same parity, then 3a +7 and 76 -4 do not. 15. Suppose x € Z. If x³ - 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, b e Z, then (a + b)³ = a³ + b³ (mod 3). 19. Let a, b, ce Z and n E N. If a = b (mod n) and a = c (mod n), then c = b (mod n). 20. If a € Z and a = 1 (mod 5), then a2 = 1 (mod 5). 21. Let a, b e Z and neN. If a = b (mod n), then a³ = 6³ (mod n). 22. Let a € Z, neN. If a has remainder r when divided by n, then a = r (mod n). 23. Let a, b eZ and n E N. 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. If n eN and 2" - 1 is prime, then n is prime. 26. If n=2k-1 for k EN, then every entry in Row n of Pascal's Triangle is odd. 27. If a = 0 (mod 4) or a = 1 (mod 4), then (2) is even. 28. If n c Z, then 4 (n²-3). T 29. If integers a and b are not both zero, then gcd(a, b) = gcd(a - b, b). 30. If a = b (mod n), then ged(a, n) = ged(b,n). 31. Suppose the division algorithm applied to a and b yields a = qb+r. Prove ged(a,b)= ged(r, b). 32. If a = b (mod n), then a and b have the same remainder when divided by n.
B. Prove the following statements using either direct or contrapositive proof. 14. If a, b e Z and a and b have the same parity, then 3a +7 and 76 -4 do not. 15. Suppose x € Z. If x³ - 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, b e Z, then (a + b)³ = a³ + b³ (mod 3). 19. Let a, b, ce Z and n E N. If a = b (mod n) and a = c (mod n), then c = b (mod n). 20. If a € Z and a = 1 (mod 5), then a2 = 1 (mod 5). 21. Let a, b e Z and neN. If a = b (mod n), then a³ = 6³ (mod n). 22. Let a € Z, neN. If a has remainder r when divided by n, then a = r (mod n). 23. Let a, b eZ and n E N. 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. If n eN and 2" - 1 is prime, then n is prime. 26. If n=2k-1 for k EN, then every entry in Row n of Pascal's Triangle is odd. 27. If a = 0 (mod 4) or a = 1 (mod 4), then (2) is even. 28. If n c Z, then 4 (n²-3). T 29. If integers a and b are not both zero, then gcd(a, b) = gcd(a - b, b). 30. If a = b (mod n), then ged(a, n) = ged(b,n). 31. Suppose the division algorithm applied to a and b yields a = qb+r. Prove ged(a,b)= ged(r, b). 32. If a = b (mod n), then a and b have the same remainder when divided by n.
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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Please detailed prove for 16,18,22,28
Transcribed Image Text:B. Prove the following statements using either direct or contrapositive proof.
14. If a, b e Z and a and b have the same parity, then 3a +7 and 76-4 do not.
15. Suppose x € Z. If x³ - 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, b e Z, then (a + b)³ = a³ + b³ (mod 3).
19. Let a, b, ce Z and n E N. If a = b (mod n) and a = c (mod n), then c = b (mod n).
20. If a € Z and a = 1 (mod 5), then a2 = 1 (mod 5).
21. Let a, b e Z and neN. If a = b (mod n), then a³ = 6³ (mod n).
22. Let a € Z, neN. If a has remainder r when divided by n, then a = r (mod n).
23. Let a, b e Z and n E N. 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. If n e N and 2" - 1 is prime, then n is prime.
26. If n=2k-1 for k EN, then every entry in Row n of Pascal's Triangle is odd.
27. If a = 0 (mod 4) or a = 1 (mod 4), then (2) is even.
28. If n c Z, then 4 (n²-3).
T
29. If integers a and b are not both zero, then gcd(a, b) = gcd(a - b, b).
30. If a = b (mod n), then ged(a, n) = ged(b,n).
31. Suppose the division algorithm applied to a and b yields a = qb+r. Prove
ged(a,b)= ged(r, b).
32. If a = b (mod n), then a and b have the same remainder when divided by n.
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