Two years hence, the ratio of the ages of ‘B’ and ‘C’ will be 5:4. The present age of ‘C’ is half of the sum of the present ages of ‘A’ and ‘B’. Three years ago, the product of the ages of ‘A’ and ‘C’ (in years) was 77. If the age of ‘B’ eight years hence from now will be ‘p’ years, then how many statement(s) among the following is/are true about ‘p’? Statement I: ‘p’ is a perfect square. Statement II: (p − 2) is a prime number. Statement III: (p + 3) is completely divisible by 6. Statement IV: (p + 5) > 28

Let the ages of ‘B’ and ‘C’ two years hence from now be ‘5k’ years and ‘4k’ years respectively. Present age of ‘B’ = (5k − 2) years Present age of ‘C’ = (4k − 2) years So, present age of ‘A’ = 2 × (4k − 2) − (5k − 2) = (3k − 2) years Now, (3k − 2 − 3) × (4k − 2 − 3) = 77 Or, (3k − 5) × (4k − 5) = 77 Or, 12k² − 15k − 20k + 25 = 77 Or, 12k² − 35k − 52 = 0 Or, (12k + 13)(k − 4) = 0 So, ‘k’ = −(13/12) or ‘k’ = 4 But ‘k’ cannot be negative. So, ‘k’ = 4 Therefore, ‘p’ = (5k − 2) + 8 = 5k + 6 = 26 Statement I 26 is not a perfect square.
So, statement I is false. Statement II 26 − 2 = 24, which is not prime.
So, statement II is false. Statement III 26 + 3 = 29, which is not completely divisible by 6.
So, statement III is false. Statement IV 26 + 5 = 31 > 28.
So, statement IV is true. Therefore, one among the four statements given is true. Hence, option E.