Among the numbers 343, 2197, 1331 and 121, which one does not belong to the group of perfect cubes?

Introduction / Context:
This odd man out question checks your familiarity with perfect cubes and basic factorization. In many aptitude tests, such problems help separate candidates who quickly recognize standard number forms from those who rely only on trial and error. The goal is to identify which number does not share the same key property as the rest.

Given Data / Assumptions:

• The four main options are 343, 2197, 1331 and 121.
• We assume that three of these are perfect cubes of integers while one is not.
• We need to find the number which breaks this perfect cube pattern.

Concept / Approach:
A perfect cube is any integer that can be written in the form n^3 where n is an integer. The direct approach is to factor each number or recall common cubes such as 7^3, 11^3 and 13^3. By matching each value to one of these known cubes, we can see which one does not fit the pattern.

Step-by-Step Solution:
Check 343: 7 * 7 * 7 = 343, so 343 = 7^3 and is a perfect cube.Check 2197: 13 * 13 * 13 = 2197, so 2197 = 13^3 and is a perfect cube.Check 1331: 11 * 11 * 11 = 1331, so 1331 = 11^3 and is a perfect cube.Check 121: 11 * 11 = 121, so 121 = 11^2 which is a perfect square, not a perfect cube.

Verification / Alternative check:
We can also verify by checking approximate cube roots. The cube root of 343 is 7 which is an integer. The cube root of 2197 is 13 which is an integer. The cube root of 1331 is 11 which is also an integer. However, the cube root of 121 is between 4 and 5 and is not an integer. Therefore, 121 fails the perfect cube test while the others pass it.

Why Other Options Are Wrong:
343 is a perfect cube (7^3) and fits the group property.
2197 is a perfect cube (13^3) and also fits the group property.
1331 is a perfect cube (11^3) and belongs to the same family.
Therefore none of these can be the odd one out, leaving only 121 as the correct choice.

Common Pitfalls:
Some learners see that 121 is a very familiar number and immediately think of it as 11^2 and may incorrectly assume all others are squares, which is not correct. Others forget the common cubes beyond 10^3 and struggle with 2197 and 1331. It is helpful to memorize cubes of integers at least up to 15 for exam purposes, as these appear frequently in number system questions.

Final Answer:
The number that is not a perfect cube is 121.