In the following aptitude question on number classification, four numbers are given (216, 125, 343 and 510). Three of these are perfect cubes, while one number is not a perfect cube. Select the odd number from the given alternatives.

Introduction / Context:
This aptitude question belongs to the odd one out and number classification topic. The options 216, 125, 343 and 510 look similar at first glance, but there is a hidden pattern based on perfect cubes. Your task is to recognise which numbers are exact cubes of integers and then pick the number that does not fit this pattern.

Given Data / Assumptions:

• Given numbers: 216, 125, 343 and 510.
• We are working with basic number properties, especially powers of integers.
• Perfect cube means a number of the form n^3 where n is an integer.
• Exactly one of the four numbers is not a perfect cube.
• No advanced mathematics is required, only familiarity with small cubes.

Concept / Approach:
The main concept tested here is recognition of perfect cubes. Many competitive exams expect you to remember cubes of small integers such as 2^3, 3^3, 4^3, 5^3, 6^3, 7^3 and so on. Once you match each option with a known cube, you can easily see which number does not match any integer cube. That non matching number will be the odd one out in the list.

Step-by-Step Solution:
Step 1: Check 216.Compute 6^3 = 6 * 6 * 6 = 216. So 216 is a perfect cube.Step 2: Check 125.Compute 5^3 = 5 * 5 * 5 = 125. So 125 is also a perfect cube.Step 3: Check 343.Compute 7^3 = 7 * 7 * 7 = 343. So 343 is again a perfect cube.Step 4: Check 510.Try nearby cubes: 7^3 = 343, 8^3 = 512, 9^3 = 729. None of these equal 510, so 510 is not a perfect cube.Step 5: Identify the odd number.Three numbers (216, 125, 343) are exact cubes of integers, but 510 is not, so 510 is the odd one out.

Verification / Alternative check:
An alternative way to check is to compare 510 with the cube 512. Since 510 is very close to 512 but not equal, and cubes are exact values, 510 cannot be any integer cube. On the other hand, you can directly confirm from memory that 5^3, 6^3 and 7^3 are 125, 216 and 343 respectively, so those three clearly qualify as perfect cubes. This confirms your identification without any doubt.

Why Other Options Are Wrong:
216: This is 6^3, so it correctly fits the perfect cube pattern.
125: This is 5^3, and therefore it also follows the same cube rule.
343: This is 7^3, matching the pattern of being a perfect cube.

Common Pitfalls:
Candidates may not memorise cubes beyond 5^3 and might hesitate with 216 and 343. Some students might also confuse 512 and 510. To avoid errors, it is helpful to know cubes at least up to 10^3, and to remember that cubes are exact. If a number is even slightly off from a known cube, it cannot be a perfect cube. Careful checking of each option prevents careless mistakes in such classification questions.

Final Answer:
The odd number is 510, because 216, 125 and 343 are perfect cubes of integers, while 510 is not a perfect cube.