Answer 6.1, 6.3, 6.4, 6.6, 6.9, 6.14, 6.15, 8.1, 8.5, 8.8, 8.13, and 8.19

Problem 6.1 Show that there exist integers m and n such that 3m + 4n = 25. Problem 6.2 Show that there is a positive integer x such that a* 4 2% + 2% 22 2 = 0. Hint: Use the rational zero test. Problem 6.3 Show that there exist two prime numbers whose product is 143. Problem 6.4 Show that there exists a point in the Cartesian system that is not on the line y=2r-3. Problem 6.5 Using a constructive proof, show that the number r = 6.32152152... is a rational number, Problem 6.6 Show that the equation x* 32% + 2o 4 = 0 has at least one solution in the interval (2,3). Problem 6.7 Use a non-constructive proof to show that there exist irrational numbers a and b such that a\" is rational. Hint: Look at the number = V2", Consider the cases is rational or is irrational. Problem 6.8 Use the method of exhaustion to show: n N, if n is even and 4 0 then =2+2 Problem 6.15 Show that for any even integer n, we have (1)" = 1. Problem 8.1 Use the proof by contradiction to prove the proposition \"There is no greatest even integer.\" Problem 8.2 Prove by contradiction that the difference of any rational number and any irrational number is irrational. Problem 8.3 Use the proof by contraposition to show that if a product of two positive real numbers is greater than 100, then at least one of the numbers is greater than 10. Problem 8.4 Use the proof by contradiction to show that the product of any nonzero rational mumber and any irrational number is irrational. Problem 8.5 Show that if n is an integer and n? is divisible by 3 then n is also divisible by 3. Problem 8.6 Show that the number /3 is irrational. Problem 8.7 Use the proof by contrapositive to show that if n and m are two integers for which n + m is even then either both n and m are even or both are odd. Problem 8.8 Use the proof by contrapositive to show that for any integer n, if 3n + 1 is even then n is odd. Problem 8.13 Prove by contradiction: Suppose that n Z. If n +5 is odd then n is even. Problem 8.14 Prove by contradiction: There exist no integers a and b such that 18a+6b = 1. Problem 8.15 Prove by contrapositive: If a and b are two integers such that a - b is not divisible by n then a and b are not divisible by n. Problem 8.16 Prove by contradiction: Suppose that a.b, and are positive real numbers. Show that if ab = then a 15 then a > 8 or b > 8