Among the numbers 343, 64, 75 and 27, identify the odd number that is not a perfect cube.

Introduction / Context:
In many aptitude and reasoning exams, you are often asked to find an odd item out from a set of numbers based on a hidden mathematical property. In this question, the hidden property is related to perfect cubes. A perfect cube is a number that can be written in the form n^3, where n is a whole number. By checking each option against this property, we can quickly see which number does not belong to the same group and therefore is the odd one out.

Given Data / Assumptions:
We are given four numbers: 343, 64, 75 and 27. We assume all numbers are positive integers. We are asked to find the number that does not share a common property with the others. The most natural property to test here is whether each number is a perfect cube.

Concept / Approach:
The key concept is the definition of a perfect cube. If a number N can be written as n^3 for some integer n, then N is a perfect cube. To solve the problem, we test each given number by seeing whether it equals the cube of an integer. If three of the numbers are perfect cubes and one is not, then the non cube is the odd one out. This is a very common pattern in odd number questions and appears frequently in competitive exams and placement tests.

Step-by-Step Solution:
Step 1: Check 343. We know 7^3 = 7 * 7 * 7 = 343, so 343 is a perfect cube. Step 2: Check 64. We know 4^3 = 4 * 4 * 4 = 64, so 64 is a perfect cube. Step 3: Check 27. We know 3^3 = 3 * 3 * 3 = 27, so 27 is a perfect cube. Step 4: Check 75. There is no integer n such that n^3 = 75. For example, 4^3 = 64 and 5^3 = 125, so 75 lies between two consecutive cubes and therefore is not a perfect cube. Step 5: Since 343, 64 and 27 are all perfect cubes but 75 is not, 75 is the number that does not fit the pattern.

Verification / Alternative check:
One quick way to verify is to look at nearby cubes. Between 4^3 = 64 and 5^3 = 125, there is no other perfect cube, so any number in that interval like 75 cannot be a cube. For the others, 3^3 = 27, 4^3 = 64 and 7^3 = 343 are well known standard cubes that appear frequently in cube tables. Because three numbers clearly match the cube pattern and one clearly does not, the conclusion is consistent.

Why Other Options Are Wrong:
343 is not the odd one out because it is exactly equal to 7^3 and is therefore a perfect cube. 64 is not the odd one out because it is equal to 4^3 and satisfies the property of being a perfect cube. 27 is not the odd one out because it is equal to 3^3 and also satisfies the same property. Thus, these three numbers form a consistent group of perfect cubes.

Common Pitfalls:
Students sometimes look at last digits only and try to guess patterns without verifying the cube relationship properly. Another mistake is to search for factors or prime factorization patterns without first checking the simple and intended property of being a perfect cube. Time pressure in exams can also cause candidates to overlook known cube values like 343 and 64, so it is useful to memorize small cubes from 1^3 to at least 10^3.

Final Answer:
Therefore, the odd number that is not a perfect cube is 75.