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      Question

      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.
      A If the data in statement I alone are sufficient, but the data in statements II and III alone are not sufficient. Correct Answer Incorrect Answer
      B If the data in statement II alone are sufficient, but the data in statements I and III alone are not sufficient. Correct Answer Incorrect Answer
      C If the data in statement III alone are sufficient, but the data in statements I and II alone are not sufficient. Correct Answer Incorrect Answer
      D If the data in any two of the three statements together are sufficient, but no single statement alone is sufficient. Correct Answer Incorrect Answer
      E If the data in all three statements together are necessary to answer the question; even any two statements together are not sufficient. Correct Answer Incorrect Answer

      Solution

      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.

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