1. Show that there exist integers m and n such that 3m + 4n = 25.

2. Show that there is a positive integer x such that x⁴ + 2x³ + x² - 2x - 2 = 0.Hint: Use the rational zero test.

3. Show that there exist two prime numbers whose product is 143.

4. Show that there exists a point in the Cartesian system that is not on the line y = 2x - 3.

5. Show that the equation x³ - 3x² + 2x - 4 = 0 has at least one solution in the interval (2, 3).

6. Use a non-constructive proof to show that there exist irrational numbers a and b such that a^b is rational. Hint: Look at the number q =√2^√2. Consider the cases q is rational or q is irrational

7. Prove the following theorem: The product of two rational numbers is a rational number.

8. Use the previous problem to prove the following corollary: The square of any rational number is rational.

9. Disprove the following statement by finding a counterexample: ∀x, y, z ∈ R,if x > y then xz > yz.

10. Disprove the following statement by finding a counterexample: ∀x ∈ R, if x > 0 then 1/(x+2) =1/x +1/2.

11. Show that the product of two odd integers is odd.

12. Identify the error in the following proof:"For all positive integer n, the product(n - 1)n(n + 2) is divisible by 3 since 3(4)(5) is divisible by 3.

13. Identify the error in the following proof:" If n and m are two different odd integers then there exists an integer k such that n = 2k + 1 and m = 2k + 1."

14. Identify the error in the following proof:"Suppose that m and n are integerssuch that n + m is even. We want to show that n - m is even. For n + mto be even, n and m must be even. Hence, n = 2k₁ and m = 2k₂ so thatn - m = 2(k₁ - k₂) which is even."