In this numerical odd-man-out question, four numbers are given: 8, 27, 100 and 125. Three of these numbers are perfect cubes of integers, whereas one number is not a perfect cube. Which number is the odd one out on the basis of the perfect cube property?

Introduction / Context:
This question belongs to the number reasoning and odd-man-out category. The numbers 8, 27, 100 and 125 are presented, and you must determine which one does not share a common mathematical property with the others. In this case, three of the numbers are perfect cubes of integers, and one is not. Recognizing common cube values is a standard requirement in many aptitude and competitive exams because it helps speed up problem solving in several topics like series, ratios and exponents.

Given Data / Assumptions:
The given numbers are 8, 27, 100 and 125.
We are looking for a property that three numbers share and one lacks.
A likely property is being a perfect cube (n^3 for some integer n).
Only one number among the four is not a perfect cube and must be chosen as the odd one out.

Concept / Approach:
A perfect cube is any integer that can be written as n^3, where n is an integer. Common cubes to remember are 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125 and so on. The simplest approach in this question is to test each number against known cube values. If three numbers match some n^3 exactly, while one does not, the unmatched number is the odd element. This strategy avoids unnecessary calculations and relies on basic memory of cube values.

Step-by-Step Solution:
Step 1: Check 8. We know that 2^3 = 2 * 2 * 2 = 8, so 8 is a perfect cube. Step 2: Check 27. We know that 3^3 = 3 * 3 * 3 = 27, so 27 is also a perfect cube. Step 3: Check 125. We know that 5^3 = 5 * 5 * 5 = 125, so 125 is a perfect cube as well. Step 4: Check 100. Nearby cubes are 4^3 = 64 and 5^3 = 125. The number 100 does not match any integer cube value between these. Step 5: Conclude that 8, 27 and 125 are perfect cubes, while 100 is not. Therefore, 100 is the odd one out.
Verification / Alternative check:
To verify, you can also think in terms of cube roots. The cube root of 8 is 2, of 27 is 3 and of 125 is 5, all integers. For 100, the cube root lies between 4 and 5 and is not an integer. Only numbers with integer cube roots are perfect cubes. This second approach confirms that 100 does not share the cube property, reinforcing your earlier conclusion.

Why Other Options Are Wrong:
8 is equal to 2^3 and is a standard small perfect cube, so it correctly matches the intended property.
27 is equal to 3^3 and belongs to the same group of perfect cubes as 8 and 125, so it is not the odd one.
125 is equal to 5^3 and again fits the perfect cube pattern, so it cannot be the odd number.
100 does not equal n^3 for any integer n and therefore is the only non-cube in the list.

Common Pitfalls:
Students sometimes confuse squares and cubes, especially with familiar numbers like 100, which is 10^2 (a perfect square) but not a perfect cube. Another pitfall is not remembering enough cube values and trying to guess. It is highly recommended to memorize cubes of integers at least from 1 to 12 for exam purposes. In odd-man-out questions, always check the most common number properties first, such as being a square, cube, prime or composite, before looking for more complicated patterns.

Final Answer:
The only number that is not a perfect cube and is therefore the odd one out is 100.