For Exercises 1-28, prove the given statement.

1. If n = 25, 100, or 169, then n is a perfect square and is a sum of two perfect squares.

2. If n is an even integer, 4 ≤ n ≤12, then n is a sum of two prime numbers.

3. For any positive integer n less than or equal to 3, n! < 2ⁿ.

4. For2≤n≤4,n 2≥ 2ⁿ.

5. The sum of even integers is even (do a direct proof).

6. The sum of even integers is even (do a proof by contradiction).

7. The sum of two odd integers is even.

8. The sum of an even integer and an odd integer is odd.

9. The product of any two consecutive integers is even.

10. The sum of an integer and its square is even.

11. The square of an even number is divisible by 4.

12. For every integer n, the number

3(n²+ 2n + 3) - 2n²

is a perfect square.

13. If a number x is positive, so is x + I (do a proof by contraposition).

14. The number n is an odd integer if and only if 3n + 5 = 6k + 8 for some integer k.

15. The number n is an even integer if and only if 3n + 2 = 6k + 2 for some integer k.

16. For x and y positive numbers, x < y if and only if X²< y².

17. If x²+2x-3=O, then x ≠2.

18. If x is an even prime number, then x = 2.

19. If two integers are each divisible by some integer n, then their sum is divisible by n.

20. If the product of two integers is not divisible by an integer n, then neither integer is divisible by n.

21. The sum of three consecutive integers is divisible by 3.

22. The square of an odd integer equals 8k + 1 for some integer k.

23. The difference of two consecutive cubes is odd.

24. The sum of the squares of two odd integers cannot be a perfect square.

25. The product of the squares of two integers is a perfect square.

26. For any two numbers x and y, |XY| = |X|Y|

27. For any two numbers x and y, |x + y|≤ |x| +|y |.

28. The value A is the average of the n numbers x₁, x₂, . x,. Prove that at least one of x₁, x₂, ... , n is greater than or equal to A.