In this number classification question, select the odd number from the following alternatives: 361, 289, 225 and 216.

Introduction / Context:
This question tests your knowledge of perfect squares and cubes. The numbers given are 361, 289, 225 and 216. Three of them are perfect squares of integers, while one number is a perfect cube rather than a square. Identifying which number does not belong to the same category helps you answer this odd one out question correctly.

Given Data / Assumptions:

• Numbers given: 361, 289, 225 and 216.
• We check whether each number is a perfect square, a perfect cube or both.
• Exactly one number should differ in nature from the others.
• We assume standard knowledge of squares such as 15^2, 17^2, 19^2 and cubes such as 6^3.

Concept / Approach:
Perfect squares are numbers that can be written as n^2, while perfect cubes can be written as n^3. To solve this question, we will try to express each number as a square of an integer. If three numbers match well known squares and one does not, that unmatched number is likely a cube. That cube number will be the odd one out because its classification differs from the other three, which are all squares.

Step-by-Step Solution:
Step 1: Check 361.19^2 = 19 * 19 = 361. So 361 is a perfect square.Step 2: Check 289.17^2 = 17 * 17 = 289. So 289 is also a perfect square.Step 3: Check 225.15^2 = 15 * 15 = 225. So 225 is a perfect square as well.Step 4: Check 216.Calculate 14^2 = 196 and 15^2 = 225, so 216 is not a perfect square between them. However, 6^3 = 6 * 6 * 6 = 216, which means 216 is a perfect cube, not a square.Step 5: Identify the odd number.Since 361, 289 and 225 are perfect squares and 216 is a perfect cube, 216 is the odd one out.

Verification / Alternative check:
You can also look at the approximate square roots. The square root of 225 is exactly 15, that of 289 is 17 and that of 361 is 19. In each case, you get an integer. The square root of 216 is between 14 and 15 and clearly not an integer. When you try small cubes, you find that 6^3 is exactly 216, confirming its nature as a cube. This difference in classification between square and cube confirms that 216 is the number that does not belong with the others.

Why Other Options Are Wrong:
361: Equal to 19^2, a perfect square, so it belongs with 289 and 225.
289: Equal to 17^2, again a perfect square consistent with the main pattern.
225: Equal to 15^2, another perfect square, not the odd one out.

Common Pitfalls:
Some students may miscalculate square values or confuse 216 with 15^2. Remembering that 15^2 = 225 and 14^2 = 196 helps avoid such confusion. It is also useful to memorize a small set of cubes such as 5^3 = 125, 6^3 = 216 and 7^3 = 343. These values frequently appear in aptitude questions and help you quickly identify whether a number is a square, a cube or neither.

Final Answer:
The odd number is 216, because 361, 289 and 225 are perfect squares, while 216 is a perfect cube.