Use the method of direct proof to prove the following statements.
1. If x is an even integer, then x^2 is even.
2. If x is an odd integer, then x^3 is odd.
3. If a is an odd integer, then a^2 + 3a + 5 is odd.
4. Suppose x, y ∈ Z. If x and y are odd, then xy is odd.
5. Suppose x, y ∈ Z. If x is even, then xy is even.
6. Suppose a, b, c ∈ Z. If a | b and a | c, then a | (b + c).
7. Suppose a, b ∈ Z. If a | b, then a^2 | b^2.
8. Suppose a is an integer. If 5 | 2a, then 5 | a.
9. Suppose a is an integer. If 7 | 4a, then 7 | a.
10. Suppose a and b are integers. If a | b, then a | (3b^3 - b^2 + 5b).
11. Suppose a, b, c, d ∈ Z. If a | b and c | d, then ac | bd.
12. If x ∈ R and 0 < x < 4, then 4/(x(4-x)) ≥ 1.
13. Suppose x, y ∈ R. If x^2 + 5y = y^2 + 5x, then x = y or x + y = 5.
14. If n ∈ Z, then 5n^2 + 3n + 7 is odd. (Try cases.)
15. If n ∈ Z, then n^2 + 3n + 4 is even. (Try cases.)
16. If two integers have the same parity, then their sum is even. (Try cases.)
17. If two integers have opposite parity, then their product is even.
18. Suppose x and y are positive real numbers. If x < y, then x^2 < y^2.
19. Suppose a, b and c are integers. If a^2 | b and b^3 | c, then a^6 | c.