All of these problems are done using generic particular.

1. Prove there are distinct integers m and n such that 1/m + 1/n is an integer.

2. There is a perfect square that can be written as the sum of two other perfect squares.

3. Disprove the following by giving a counterexample: For all integers n, if n is odd, then (n 1)/2 is odd.

4. Disprove the following by giving a counterexample: For all integers m and n, if 2m + n is odd then m and n are both odd.

5. Prove the following by the method of exhaustion: For each integer n with 1 n 10, n^2 n + 11 is a prime number.

6. By using the properties of even and odd integers (page 167 in your book), show that the product of any even integer and any integer is even.

7. Represent the following recurring decimals as rational numbers:

(a) 43.738738738738 . . .

(b) 32.456745674567 . . .

(c) 56.78311311311311 . . .

(d) 21.134191191191191 . . .

8. Using the properties of even and odd integers (page 167 in your book) answer the following: True or False? If k is any even ingeger and m is any odd integer, then (k + 2)^2 (m 1)^2 is even. Explain.

9. Using Theorem 4.2.1 (Page 165 in your book) and the fact that the sum, difference, product of any two rational numbers is rational, argue that (5s^3) + (8s^2) 10 is rational when s is rational number.

10. Argue that for all real numbers a and b, if a < b, then a < (a + b)/2 < b. (Give a convining argument - no need to use generic particular). Use this to show that given any two rational numbers r and s with r < s, there ia another rational number between r and s.