Section 2.1 Proof Techniques
109
27. For x and y positive numbers, x < y if and only if x
2
< y
2
.
28. If x
2
+ 2x − 3 = 0, then x ∙ 2.
29. If n is an even prime number, then n = 2.
30. The sum of three consecutive integers is divisible by 3.
31. If two integers are each divisible by some integer n, then their sum is divisible by n.
32. If the product of two integers is not divisible by an integer n, then neither integer is divisible by n.
33. If n, m, and p are integers and n 0 m and m 0 p, then n 0 p.
34. If n, m, p, and q are integers and n 0 p and m 0 q, then nm 0 pq.
35. The square of an odd integer equals 8k + 1 for some integer k.
36. The sum of the squares of two odd integers cannot be a perfect square. (Hint: Use Exercise 35.)
37. The product of the squares of two integers is a perfect square.
38. The difference of two consecutive cubes is odd.
39. For any two numbers x and y, 0 x + y 0 ≤ 0 x 0 + 0 y 0 .
40. For any two numbers x and y, 0 xy 0 = 0 x 0 0 y 0 .
41. The value A is the average of the n numbers x 1 , x 2 , … , x n . Prove that at least one of x 1 , x 2 , … , x n is greater
than or equal to A.
42. Suppose you were to use the steps of Example 11 to attempt to prove that "4 is not a rational number. At
what point would the proof not be valid?
43. Prove that "3 is not a rational number.
44. Prove that "5 is not a rational number.
45. Prove that "
3
2 is not a rational number.
46. Prove that log 2 5 is not a rational number (log 2 5 = x means 2
x
= 5).
For Exercises 47–72, prove or disprove the given statement.
47. 0 is an even number.
48. 91 is a composite number.
49. 297 is a composite number.
50. 83 is a composite number.
51. The difference between two odd integers is odd.
52. The difference between two even integers is even.
53. The product of any three consecutive integers is even.
54. The sum of any three consecutive integers is even.
55. The sum of an integer and its cube is even.
56. The number n is an even integer if and only if n
3
+ 13 is odd.
57. The product of an integer and its square is even.
58. Any positive integer can be written as the sum of the squares of two integers.
59. The sum of the square of an odd integer and the square of an even integer is odd.
60. If n is a positive integer that is a perfect square, then n + 2 is not a perfect square.
61. For a positive integer n, n +
1
n
≥ 2.
109
27. For x and y positive numbers, x < y if and only if x
2
< y
2
.
28. If x
2
+ 2x − 3 = 0, then x ∙ 2.
29. If n is an even prime number, then n = 2.
30. The sum of three consecutive integers is divisible by 3.
31. If two integers are each divisible by some integer n, then their sum is divisible by n.
32. If the product of two integers is not divisible by an integer n, then neither integer is divisible by n.
33. If n, m, and p are integers and n 0 m and m 0 p, then n 0 p.
34. If n, m, p, and q are integers and n 0 p and m 0 q, then nm 0 pq.
35. The square of an odd integer equals 8k + 1 for some integer k.
36. The sum of the squares of two odd integers cannot be a perfect square. (Hint: Use Exercise 35.)
37. The product of the squares of two integers is a perfect square.
38. The difference of two consecutive cubes is odd.
39. For any two numbers x and y, 0 x + y 0 ≤ 0 x 0 + 0 y 0 .
40. For any two numbers x and y, 0 xy 0 = 0 x 0 0 y 0 .
41. The value A is the average of the n numbers x 1 , x 2 , … , x n . Prove that at least one of x 1 , x 2 , … , x n is greater
than or equal to A.
42. Suppose you were to use the steps of Example 11 to attempt to prove that "4 is not a rational number. At
what point would the proof not be valid?
43. Prove that "3 is not a rational number.
44. Prove that "5 is not a rational number.
45. Prove that "
3
2 is not a rational number.
46. Prove that log 2 5 is not a rational number (log 2 5 = x means 2
x
= 5).
For Exercises 47–72, prove or disprove the given statement.
47. 0 is an even number.
48. 91 is a composite number.
49. 297 is a composite number.
50. 83 is a composite number.
51. The difference between two odd integers is odd.
52. The difference between two even integers is even.
53. The product of any three consecutive integers is even.
54. The sum of any three consecutive integers is even.
55. The sum of an integer and its cube is even.
56. The number n is an even integer if and only if n
3
+ 13 is odd.
57. The product of an integer and its square is even.
58. Any positive integer can be written as the sum of the squares of two integers.
59. The sum of the square of an odd integer and the square of an even integer is odd.
60. If n is a positive integer that is a perfect square, then n + 2 is not a perfect square.
61. For a positive integer n, n +
1
n
≥ 2.