In this aptitude question, you are asked to find the odd number from the following set of alternatives, where three of the numbers are perfect cubes and one number does not follow that perfect cube pattern: 27, 8, 64 and 124.

Introduction / Context:
This question is a classic odd one out problem from the aptitude section. You are given four numbers and you must identify which number does not share the same mathematical property as the others. Specifically, this problem focuses on the concept of perfect cubes, which is very common in number system questions in competitive examinations and reasoning tests.

Given Data / Assumptions:

• The given numbers are 27, 8, 64 and 124.
• All numbers are positive integers.
• We assume that the intended pattern relates to basic powers of integers such as squares or cubes.

Concept / Approach:
The main idea is to test whether each number is a perfect cube. A perfect cube is an integer that can be expressed as n^3 for some integer n. If three numbers are perfect cubes and one is not, then the non cube number will be the odd one out. We do not need advanced number theory here, only familiarity with basic cubes of small integers like 2, 3, 4 and 5.

Step-by-Step Solution:
Step 1: Check 27. We know 3^3 = 27, so 27 is a perfect cube.Step 2: Check 8. We know 2^3 = 8, so 8 is a perfect cube.Step 3: Check 64. We know 4^3 = 64, so 64 is a perfect cube.Step 4: Check 124. There is no integer n such that n^3 = 124. We know 4^3 = 64 and 5^3 = 125, so 124 lies between 4^3 and 5^3 and therefore it is not a perfect cube.Step 5: Since three numbers are perfect cubes and one number is not, the non cube must be the odd one out.
Verification / Alternative check:
As another quick check, list nearby cubes: 2^3 = 8, 3^3 = 27, 4^3 = 64 and 5^3 = 125. We see that 124 is just one less than 125, so it cannot be a perfect cube. This confirms that 124 is the only number that fails the perfect cube test among the given options.

Why Other Options Are Wrong:
27 is wrong as the odd one out because it is a perfect cube, equal to 3^3, so it fits the pattern.
8 is wrong because it is also a perfect cube, equal to 2^3, so it matches the pattern of the majority of the numbers.
64 is wrong as the odd one out because it is again a perfect cube, equal to 4^3, and therefore follows the same property as 8 and 27.

Common Pitfalls:
Many learners may quickly focus only on the size of the numbers or their even and odd nature rather than checking powers. For example, they might mistakenly think that 27 is different because it is the only odd number, but the actual intended pattern is about perfect cubes. Ignoring mathematical properties like squares and cubes is a frequent mistake in number based reasoning questions. It is also possible for someone to confuse 124 with 125 and assume it is a cube, which is not correct.

Final Answer:
The only number that is not a perfect cube is 124, so 124 is the correct odd one out.