Question

B. Prove the following statements using either direct or contrapositive proof. Sometimes one approach will be much easier than the other. 14. If a, b ? Z and a and b have the same parity, then 3a + 7 and 7b - 4 do not. 15. Suppose x ? Z. If x^3 - 1 is even, then x is odd. 16. Suppose x ? Z. If x + y is even, then x and y have the same parity. 17. If n is odd, then 8 | (n^2 - 1). 18. For any a, b ? Z, it follows that (a + b)^3 ? a^3 + b^3 (mod 3). 19. Let a, b ? Z and n ? 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 a^2 ? 1 (mod 5). 21. Let a, b ? Z and n ? N. If a ? b (mod n), then a^3 ? b^3 (mod n). 22. Let a ? Z, n ? N. If a has remainder r when divided by n, then a ? r (mod n). 23. Let a, b, c ? Z and n ? N. If a ? b (mod n), then ca ? cb (mod n). 24. If a ? b (mod n) and c ? d (mod n), then ac ? bd (mod n). 25. If n ? N and 2^n - 1 is prime, then n is prime. 26. If n = 2^k - 1 for k ? N, then every entry in Row n of Pascal's Triangle is odd. 27. If a ? 0 (mod 4) or a ? 1 (mod 4), then (a2) is even. 28. If n ? Z, then 4 ? (n^2 - 3). 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 gcd(a, n) = gcd(b, n). 31. Suppose the division algorithm applied to a and b yields a = qb + r. Then gcd(a, b) = gcd(r, b).

          B. Prove the following statements using either direct or contrapositive proof. Sometimes one approach will be much easier than the other.
14. If a, b ? Z and a and b have the same parity, then 3a + 7 and 7b - 4 do not.
15. Suppose x ? Z. If x^3 - 1 is even, then x is odd.
16. Suppose x ? Z. If x + y is even, then x and y have the same parity.
17. If n is odd, then 8 | (n^2 - 1).
18. For any a, b ? Z, it follows that (a + b)^3 ? a^3 + b^3 (mod 3).
19. Let a, b ? Z and n ? 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 a^2 ? 1 (mod 5).
21. Let a, b ? Z and n ? N. If a ? b (mod n), then a^3 ? b^3 (mod n).
22. Let a ? Z, n ? N. If a has remainder r when divided by n, then a ? r (mod n).
23. Let a, b, c ? Z and n ? N. If a ? b (mod n), then ca ? cb (mod n).
24. If a ? b (mod n) and c ? d (mod n), then ac ? bd (mod n).
25. If n ? N and 2^n - 1 is prime, then n is prime.
26. If n = 2^k - 1 for k ? N, then every entry in Row n of Pascal's Triangle is odd.
27. If a ? 0 (mod 4) or a ? 1 (mod 4), then (a2) is even.
28. If n ? Z, then 4 ? (n^2 - 3).
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 gcd(a, n) = gcd(b, n).
31. Suppose the division algorithm applied to a and b yields a = qb + r. Then gcd(a, b) = gcd(r, b).

B. Prove the following statements using either direct or contrapositive proof. Sometimes one approach will be much easier than the other.
14. If a, b ? Z and a and b have the same parity, then 3a + 7 and 7b - 4 do not.
15. Suppose x ? Z. If x^3 - 1 is even, then x is odd.
16. Suppose x ? Z. If x + y is even, then x and y have the same parity.
17. If n is odd, then 8 | (n^2 - 1).
18. For any a, b ? Z, it follows that (a + b)^3 ? a^3 + b^3 (mod 3).
19. Let a, b ? Z and n ? 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 a^2 ? 1 (mod 5).
21. Let a, b ? Z and n ? N. If a ? b (mod n), then a^3 ? b^3 (mod n).
22. Let a ? Z, n ? N. If a has remainder r when divided by n, then a ? r (mod n).
23. Let a, b, c ? Z and n ? N. If a ? b (mod n), then ca ? cb (mod n).
24. If a ? b (mod n) and c ? d (mod n), then ac ? bd (mod n).
25. If n ? N and 2^n - 1 is prime, then n is prime.
26. If n = 2^k - 1 for k ? N, then every entry in Row n of Pascal's Triangle is odd.
27. If a ? 0 (mod 4) or a ? 1 (mod 4), then (a2) is even.
28. If n ? Z, then 4 ? (n^2 - 3).
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 gcd(a, n) = gcd(b, n).
31. Suppose the division algorithm applied to a and b yields a = qb + r. Then gcd(a, b) = gcd(r, b).

Added by Monique V.

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