In this number classification question on perfect cubes, select the odd number from the following alternatives: 2197, 3375, 4099 and 2744.

Introduction / Context:
This aptitude question focuses on classification of numbers based on perfect cubes. You are given four numbers 2197, 3375, 4099 and 2744 and asked to identify the odd number out. Three of the numbers are exact cubes of integers, while one number is not a perfect cube. Recognizing standard cubes helps to quickly solve this kind of problem in competitive exams.

Given Data / Assumptions:

• Numbers given: 2197, 3375, 4099 and 2744.
• We need to check whether each number is an exact cube of some integer.
• Exactly one of these numbers is not a perfect cube.
• Only basic knowledge of common cubes is required, for example 13^3, 14^3 and 15^3.

Concept / Approach:
A perfect cube is a number that can be written as n^3 for some integer n. To solve the question, we test each number to see if it is a cube of an integer. If we recall common cubes from memory or compute them quickly, we can match three of the numbers to exact cubes and see which one does not fit. That unmatched number will be the odd one out in the group.

Step-by-Step Solution:
Step 1: Check 2197.Compute 13^3 = 13 * 13 * 13 = 169 * 13 = 2197. So 2197 is a perfect cube, specifically 13^3.Step 2: Check 3375.Compute 15^3 = 15 * 15 * 15 = 225 * 15 = 3375. So 3375 is also a perfect cube, equal to 15^3.Step 3: Check 2744.Compute 14^3 = 14 * 14 * 14 = 196 * 14 = 2744. So 2744 is a perfect cube as well, namely 14^3.Step 4: Check 4099.There is no nearby integer whose cube equals 4099. For example, 16^3 = 4096 and 17^3 = 4913. 4099 lies between these and therefore cannot be written as n^3 for any integer n.Step 5: Identify the odd number.Since 2197, 3375 and 2744 are perfect cubes but 4099 is not, 4099 is the odd number out.

Verification / Alternative check:
A quick alternative check is to note that 4099 is only three more than 4096, which is 16^3. If a number is an exact cube, it will match n^3 for some integer n, not sit in between the cubes of consecutive integers. Since 4099 is strictly between 16^3 and 17^3, there is no integer whose cube equals 4099. This confirms that 4099 is not a perfect cube, while the other three match exactly with 13^3, 14^3 and 15^3 respectively.

Why Other Options Are Wrong:
2197: Equal to 13^3, so it is a perfect cube and not the odd one.
3375: Equal to 15^3, again a perfect cube and consistent with 2197 and 2744.
2744: Equal to 14^3, another perfect cube belonging to the same group.

Common Pitfalls:
Students sometimes misremember cube values or confuse them with squares. For numbers in the 2000 to 5000 range, it is very useful to know that 13^3 = 2197, 14^3 = 2744, 15^3 = 3375 and 16^3 = 4096. Remembering a few such key cubes can greatly speed up calculations in many exam problems, especially when identifying perfect cubes or working with volume formulas based on cube dimensions.

Final Answer:
The odd number out is 4099, because 2197, 3375 and 2744 are all perfect cubes of integers, whereas 4099 is not a perfect cube.