Question

‘a’ and ‘b’ are two natural numbers. Is ‘b’ a perfect square?
Statement 1: ‘b’ is divisible by (a + 1)2
Statement 2: \(b < 100\)

• A. statement (1) alone is sufficient to answer the question
• B. statement (2) alone is sufficient to answer the question
• C. both the statements together are needed to answer the question
• D. statement (1) alone or statement (2) alone is sufficient to answer the question
• E. neither statement (1) nor statement (2) suffices to answer the question

Answer 1

Answer: (E)

The correct answer is option (E):

neither statement (1) nor statement (2) suffices to answer the question

Let's analyze the problem. We want to determine if 'b' is a perfect square, given that 'a' and 'b' are natural numbers.

Statement 1: 'b' is divisible by (a + 1)^2

This statement tells us that 'b' is a multiple of (a + 1)^2. However, it doesn't guarantee that 'b' is a perfect square. For example, if a = 1, then (a + 1)^2 = 4. 'b' could be 4 (a perfect square) or 8, 12, 16 (a perfect square), 20, etc. Since 'b' can be a perfect square or not, statement 1 alone isn't sufficient.

Statement 2: b < 100

This statement limits the possible values of 'b'. If we consider only this statement, 'b' could be any natural number less than 100. It could be a perfect square (1, 4, 9, 16, 25, 36, 49, 64, 81) or not (2, 3, 5, etc.). Therefore, statement 2 alone isn't sufficient.

Now, let's consider both statements together:

We know b < 100 and b is divisible by (a + 1)^2.
If a = 1, then (a + 1)^2 = 4. 'b' could be 4, 16, 36, 64 - all perfect squares. However, it could also be 8, 12, 20, etc., but these possibilities are limited to those less than 100.
If a = 2, then (a + 1)^2 = 9. 'b' could be 9, 36, 81 - all perfect squares. But b could also be 18, 27, 45, etc., but those are less than 100.
If a = 3, then (a + 1)^2 = 16. 'b' could be 16, 64 - all perfect squares. But it could also be 32, 48, 80, etc.
If a = 4, then (a + 1)^2 = 25. b could be 25, 50, 75. Only 25 is a perfect square.
If a = 5, then (a + 1)^2 = 36. b could be 36, 72. Only 36 is a perfect square.
If a = 6, then (a + 1)^2 = 49. b could be 49. It is a perfect square.
If a = 7, then (a + 1)^2 = 64. b could be 64. It is a perfect square.
If a = 8, then (a + 1)^2 = 81. b could be 81. It is a perfect square.
If a = 9, then (a + 1)^2 = 100 which is not possible since b < 100.

Even with both statements, 'b' can be a perfect square or not. So, the statements together are insufficient.

Therefore, the correct answer is: neither statement (1) nor statement (2) suffices to answer the question.