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)

To determine whether 'b' is a perfect square, let's analyze the given statements individually and collectively. 

1. Statement 1: \(b\) is divisible by \((a + 1)^2\).
   • This implies that \(b\) can be expressed as \(b = k \cdot (a + 1)^2\) where \(k\) is some integer. While \((a + 1)^2\) is a perfect square, it does not ensure that \(k\) is also a perfect square.
   • Thus, \(b\) being divisible by \((a + 1)^2\) does not guarantee \(b\) is a perfect square. For instance, if \(a = 2\) and \(k = 2\), then \(b = 2 \cdot (2 + 1)^2 = 18\), which is not a perfect square.
2. Statement 2: \(b < 100\).
   • This statement alone provides a bound on the possible values of \(b\), but it does not provide any specific information about \(b\) being a perfect square.
   • While there are several perfect squares less than 100 (like 1, 4, 9, ..., 81), this alone cannot determine if \(b\) indeed is a perfect square.
3. Combining both statements:
   • Even after combining both statements, we cannot definitively state if \(b\) is a perfect square. For instance, using statement 1, \(b = 18\) is possible, which satisfies the condition \({(a + 1)^2 = 9, k = 2}\) and also \(b < 100\), yet 18 is not a perfect square.

Therefore, neither statement 1 nor statement 2 alone, nor both combined, are sufficient to conclusively determine whether \(b\) is a perfect square.

Conclusion: The correct answer is that neither statement (1) nor statement (2) suffices to answer the question.