Average of five two-digit consecutive even natural numbers is (m + 9), and the average of five two-digit consecutive odd natural numbers is (n - 6). When each set is arranged in increasing order, sum of the fourth even number and the second odd number is 59. The fifth odd number is three less than the second even number.
Which of the following statement(s) is/are true?
I. The difference between the fourth even number and the first odd number is 15.
II. The third odd number is a perfect square.
III. The first even number is greater than 27. 

Five even numbers: a, (a + 2) , (a + 4) , (a + 6) , (a + 8) Five odd numbers: b, (b + 2) , (b + 4) , (b + 6) , (b + 8) Given, (a + 6) + (b + 2) = 59 Or, a + b = 51 ------ (I) b + 8 = a + 2 - 3 Or, a – b = 9 --------- (II) On adding equation I and II, We get, a + b + a – b = 51 + 9 Or, 2a = 60 Or, 'a' = 30 On putting value of 'a' in equation II, We get, 'b' = 30 - 9 = 21 So, the five even numbers: 30, 32, 34, 36, 38

And the five odd numbers: 21, 23, 25, 27, 29

Statement I:

Required difference = 36 - 21 = 15

So, statement I is true.

Statement II:

Third odd number = 25, which is a perfect square.

So, statement II is true.

Statement III:

First even number = 30, and 30 > 27.

So, statement III is true.

Therefore, all three statements I, II and III are true.