In the following question, four numbers are given. Three of them are perfect squares and one is not. Select the number which is not a perfect square (odd number out).

Introduction / Context:
This odd one out question is based on the concept of perfect squares. It is a common type of problem in quantitative aptitude sections, where the candidate must recognise numbers that can be written as the square of an integer. Identifying perfect squares quickly is useful in many topics, such as simplifying roots, solving equations, and working with geometric areas.

Given Data / Assumptions:
- The options are 169, 421, 529, 289, and 361.
- A perfect square is an integer that can be expressed as n^2 for some integer n.
- It is given or implied that three of the numbers are perfect squares and one is not.
- We must select the number that is not a perfect square as the odd one out.

Concept / Approach:
The approach is to test each number to see whether it is a perfect square. For three digit numbers, we can check nearby squares such as 10^2 = 100, 11^2 = 121, up to about 20^2 = 400 and slightly beyond. If a number matches one of these known squares, it is a perfect square. If it does not, then it is not a perfect square. The single number that fails this test will be our answer.

Step-by-Step Solution:
Step 1: Check 169. 13^2 = 169, so 169 is a perfect square.Step 2: Check 529. 23^2 = 529, so 529 is a perfect square.Step 3: Check 289. 17^2 = 289, so 289 is a perfect square.Step 4: Check 361. 19^2 = 361, so 361 is also a perfect square.Step 5: Now check 421. The square of 20 is 400 and the square of 21 is 441. 421 lies between 400 and 441 but is not equal to either of them.Step 6: Therefore, there is no integer n such that n^2 = 421, so 421 is not a perfect square and is the odd one out.

Verification / Alternative Check:
We can verify again by examining the differences from nearby squares. 421 - 400 = 21 and 441 - 421 = 20, which shows that 421 is in between two consecutive perfect squares. A number that lies strictly between consecutive perfect squares cannot itself be a square. On the other hand, 169, 289, 361, and 529 match exactly with 13^2, 17^2, 19^2, and 23^2 respectively. This confirms that the pattern is correct: four options are perfect squares and 421 is not.

Why Other Options Are Wrong:
169 is not the odd one out because it is the square of 13 and clearly a perfect square. 529 is also not odd because it is the square of 23. 289 is a perfect square as 17^2, so it fits the main group. 361 is equal to 19^2, so it also belongs with the other perfect squares. Since these four all have integer square roots, none of them can be chosen as the odd one out.

Common Pitfalls:
Sometimes learners may think that 421 looks special because of its digits and pick it without checking the square property, which is just a guess. Others may miscalculate squares of 19 or 23 and become uncertain about 361 or 529. To avoid such mistakes, it is helpful to memorise squares of integers at least up to 25 and to practice verifying by checking the range between consecutive squares. Doing so makes odd one out questions based on perfect squares straightforward and fast to solve.

Final Answer:
421