Proof

1. Prove that for all .
2. By finding counter-examples, disprove the following statements:
   (a) If is a rational number and , then and are both rational numbers.
   (b) If then is a prime number.
3. Prove by exhaustion that if then is a prime number.
4. Prove by contradiction that the product of a rational number and an irrational number is irrational.
5. Prove that when the square of an odd number is divided by 8 , the remainder is 1 .
6. Prove that the final digit of a square number cannot be 3 .
7. Prove by contradiction that is irrational.
8. (a) Prove by contradiction that is irrational.
   (b) Explain where the proof above fails if you try to use it to prove that is irrational.