Discrete Mathematics : Ten Proofs

Prove the given statement:

1) The sum of an integer and it's square is even.

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

3) The sum of any three consecutive integers is even.

4) The product of two rational numbers is rational.

5) The product of two irrational numbers is irrational.

6) Prove that the cube root of 2 is not a rational number.

7) Prove that the square root of 3 is not a rational number.

8) If n is an even prime number, then n=2.

9) If n is an even integer, 4 is less than or equal to n which is less than or equal to 12, then n is a sum of two prime numbers.

10) For every integer n, the number 3(n^2 + 2n + 3) - 2n^2  is a perfect square.