Each question is followed by three statements I, II and
III. You have to decide whether the data given in the statements are sufficient to answer the question. What is the value of the two-digit integer N?
I. N is a multiple of 8.
II. N is a perfect square.
III. N is greater than 50.

ATQ, Given N is a two-digit positive integer. From I: Multiples of 8 between 10 and 99: 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 ⇒ Many possibilities ⇒ I alone not sufficient. From II: Two-digit perfect squares: 16, 25, 36, 49, 64, 81 ⇒ Many possibilities ⇒ II alone not sufficient. From III: N > 50, two-digit ⇒ 51, 52, …, 99 ⇒ Many possibilities ⇒ III alone not sufficient. Now check combinations of two: I + II: Numbers that are both multiples of 8 and perfect squares: Among {16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96} and {16, 25, 36, 49, 64, 81}: Common: 16, 64 So N could be 16 or 64 ⇒ not unique. I + II not sufficient. I + III: Multiples of 8 greater than 50: 56, 64, 72, 80, 88, 96 More than one ⇒ not sufficient. II + III: Perfect squares greater than 50: 64, 81 More than one ⇒ not sufficient. All three I + II + III: From I + II, N ∈ {16, 64} From III, N > 50 ⇒ N = 64 only. So N is uniquely determined as 64. Only all three statements together are necessary and sufficient.