In the following question, four numbers are given. Three of them have a digit sum equal to 9, while one number has a different digit sum. Select the number whose digit sum is not equal to 9 (odd number out).

Introduction / Context:
This is a number based reasoning question that uses the idea of digit sums. In many aptitude problems, examiners hide patterns in the sum of digits of numbers rather than in the numbers themselves. Here, three numbers share a common digit sum and one does not. Your task is to detect this hidden rule and identify the number that breaks it.

Given Data / Assumptions:
- The options are 243, 264, 333, 405, and 522.
- The digit sum of a number is obtained by adding all its digits together.
- In most options, the digit sum equals 9.
- Exactly one number has a digit sum different from 9, making it the odd one out.

Concept / Approach:
The main approach is to compute the sum of digits for each number and then compare the results. Digit sum is a useful tool because it often reveals divisibility by 3 or 9 and can show hidden regularities across several numbers. Once you know the common digit sum, you can quickly spot which number does not fit this pattern.

Step-by-Step Solution:
Step 1: For 243, digit sum is 2 + 4 + 3 = 9.Step 2: For 333, digit sum is 3 + 3 + 3 = 9.Step 3: For 405, digit sum is 4 + 0 + 5 = 9.Step 4: For 522, digit sum is 5 + 2 + 2 = 9.Step 5: For 264, digit sum is 2 + 6 + 4 = 12, which is not equal to 9.Step 6: Since all numbers except 264 have digit sum 9, 264 is the only one that breaks the pattern.

Verification / Alternative check:
Digit sum 9 also implies divisibility by 9. So 243, 333, 405, and 522 are all divisible by 9. Checking 264, we see that 264 divided by 9 does not give an integer, because 9 * 29 = 261 and 9 * 30 = 270. The remainder confirms that 264 is not a multiple of 9. This alternative view supports our earlier conclusion that 264 is the odd one out based on the digit sum rule.

Why Other Options Are Wrong:
243, 333, 405, and 522 all share the common property that their digits add up to 9. This automatically links them through the same divisibility rule. Because they follow the same pattern, these four numbers belong to one group and cannot be selected as the odd number in this question.

Common Pitfalls:
A common mistake is to look for more complicated algebraic relationships between the numbers instead of starting with simple checks like digit sums and divisibility. Another error is miscalculating the sums under exam pressure. Practising digit sum operations improves speed and accuracy, which helps in many other questions involving divisibility by 3 and 9 as well.

Final Answer:
264