7. (8pts) Let n be a positive integer. Use a direct proof to show that "If n has odd number of divisors, n is a perfect square." (Note that a positive interger a is a perfect square if a=b2 for some bZ.) 8. (8pts) Prove that "Remainder of square of an odd positive integer divided by 8 is 1 ". Hint: note that one of any two consecutive integers is even. 9. (20pts) A rational number is a number that can be expressed in the form ba where a and b are integers and b is not 0 . An irrational number is a number that cannot be expressed in the form ba for any integers a and b. Prove or disprove the following: (a) 7 is an irrational number. (b) The product of two irrational numbers is irrational. (c) The sum of two irrational numbers is irrational. (d) If a is a rational number then its inverse a1 is a rational number. (e) If a is an irrational number then its inverse a1 is an irrational number. 10. (8pts) Given a positive integer n prove the following: " n is even if and only if 13n+8 is even