In this aptitude question on odd numbers, select the number that does not follow the same property as the others: 132, 145, 187 and 144.

Introduction / Context:
Numerical odd one out questions check whether you can quickly classify numbers based on mathematical properties such as being a perfect square, perfect cube, prime number or composite number. In this problem, you are given four numbers and asked to find the one which does not share the common characteristic that the remaining three have. Such questions are frequent in bank exams, SSC exams and other competitive tests.

Given Data / Assumptions:

• The four alternatives are: 132, 145, 187 and 144.
• We are asked to select the odd number from the given alternatives.
• We assume standard school level number theory concepts like squares and cubes.

Concept / Approach:
A natural approach is to check if any of the numbers is a perfect square, perfect cube or prime. Often, only one number will have a special property while the others are ordinary composite numbers. Here, 144 is immediately familiar because it appears in square tables. We will test whether 144 is a perfect square and whether the other numbers are squares as well.

Step-by-Step Solution:
Step 1: Check 144. We know that 12 * 12 = 144, so 144 is a perfect square. Step 2: Check 132. The square of 11 is 121 and the square of 12 is 144. Since 132 lies between 121 and 144, it cannot be a perfect square. Step 3: Check 145. Again, 144 is 12^2 and 169 is 13^2. 145 is between these two squares, so it is not a perfect square. Step 4: Check 187. The squares near it are 169 (13^2) and 196 (14^2). Since 187 is between them, it is not a perfect square. Step 5: Therefore, only 144 is a perfect square, while 132, 145 and 187 are not. That makes 144 the odd one out.

Verification / Alternative check:
Another way to verify is to factorise the numbers. 144 factors into 2 * 2 * 2 * 2 * 3 * 3, which can be grouped into equal pairs of prime factors, confirming it is a perfect square. The other numbers do not factorise into completely paired primes, so they are not squares. This reinforces our conclusion that 144 has a special property which the others lack.

Why Other Options Are Wrong:
132 is wrong as the odd one out because it is not a perfect square and behaves similarly to 145 and 187 in this respect. 145 is wrong as the odd one out because, like 132 and 187, it lies between two consecutive squares and is not itself a square. 187 is wrong as the odd one out because it is another non square composite number without the special square property.

Common Pitfalls:
A common mistake is to focus only on parity, that is, whether a number is odd or even. In this set, 132 and 144 are even while 145 and 187 are odd, which does not give a clear majority pattern. Another pitfall is looking for complicated divisibility relations when a simple and well known property like being a perfect square is sufficient. Always start by testing basic patterns such as squares, cubes and primes before attempting advanced ideas.

Final Answer:
The only perfect square among the given numbers, and therefore the odd one out, is 144.