Find the odd number from the given alternatives: 24, 49, 64 and 81, where three of these are perfect squares and one is not.

Introduction / Context:
This reasoning question asks you to find the odd number out among four given options. The numbers are 24, 49, 64 and 81, labelled as options A, B, C and D. In this type of question, three items follow a common numerical pattern or property, while one item does not. The task is to detect the pattern and identify which number does not fit that pattern.

Given Data / Assumptions:

• The four numbers given are 24, 49, 64 and 81.
• The options are labelled as A for 24, B for 49, C for 64 and D for 81.
• We are looking for a common mathematical property that three numbers share.
• The remaining number, which does not share this property, will be the odd one out.
• We assume usual properties of integers such as squares and factors.

Concept / Approach:
A common approach in such odd one out questions is to test for simple number properties such as parity, divisibility or being a perfect square. Looking at 49, 64 and 81, we can notice that 7 * 7 = 49, 8 * 8 = 64 and 9 * 9 = 81. This suggests that these three are perfect squares. We then check whether 24 can be written as n * n for some integer n. If it cannot, then 24 is the odd one out because it is not a perfect square while the others are.

Step-by-Step Solution:
Step 1: Check 49. It is equal to 7 * 7, so 49 is a perfect square.Step 2: Check 64. It is equal to 8 * 8, so 64 is a perfect square.Step 3: Check 81. It is equal to 9 * 9, so 81 is a perfect square.Step 4: Check 24. There is no whole number n such that n * n = 24, because 4 * 4 = 16 and 5 * 5 = 25, so 24 lies between these two squares.Step 5: Therefore 24 is not a perfect square, while 49, 64 and 81 are all perfect squares of consecutive integers.

Verification / Alternative check:
You can also verify this by listing small perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81 and so on. In this list 49, 64 and 81 clearly appear. The number 24 never appears, which confirms that it is not a perfect square. Since the pattern that three numbers share is that they are squares of 7, 8 and 9, the remaining number 24 must be the one that breaks the pattern.

Why Other Options Are Wrong:
Option B corresponds to 49, which is 7 squared and fits the perfect square pattern. Option C corresponds to 64, equal to 8 squared, again fitting the pattern. Option D corresponds to 81, which is 9 squared and also fits the pattern. Only option A, representing 24, does not follow this rule, so the other options are not correct as the odd one out.

Common Pitfalls:
Sometimes students focus on less useful properties, such as whether the numbers are even or odd, or whether they are divisible by some small prime. While 24 and 64 are even, 49 and 81 are odd, which does not isolate a single odd one out. The stronger and more consistent pattern is the perfect square property. Always check for common simple patterns like squares and cubes before moving to more complicated ideas.

Final Answer:
The odd number out is 24, which corresponds to option A.