In the following question, four four digit numbers are given. In three of these numbers, the sum of the digits is a multiple of 9, while in one number the sum of the digits is not a multiple of 9. Select the number whose digit sum is not divisible by 9 (odd number out).

Introduction / Context:
This number reasoning question uses the concept of digit sums and divisibility by 9. You are asked to find which four digit number behaves differently from the others when you apply the rule for multiples of 9. This type of question is common in aptitude tests because it checks whether you know and can quickly apply divisibility rules without long division.

Given Data / Assumptions:
- The options are 2691, 9900, 5632, 2790, and 4509.
- For divisibility by 9, a number is a multiple of 9 if the sum of its digits is a multiple of 9.
- In most options, the sum of the digits equals 18, which is a multiple of 9.
- Exactly one option has a digit sum that is not a multiple of 9, making it the odd one out.

Concept / Approach:
The approach is to calculate the digit sum for each four digit number and then check whether that sum is divisible by 9. When the digit sum is 9, 18, 27, or any other multiple of 9, the original number is divisible by 9. The single number that does not satisfy this condition will be the answer. This simple procedure is much faster than performing full division for each option.

Step-by-Step Solution:
Step 1: For 2691, digit sum is 2 + 6 + 9 + 1 = 18, which is divisible by 9.Step 2: For 9900, digit sum is 9 + 9 + 0 + 0 = 18, again divisible by 9.Step 3: For 2790, digit sum is 2 + 7 + 9 + 0 = 18, also a multiple of 9.Step 4: For 4509, digit sum is 4 + 5 + 0 + 9 = 18, again a multiple of 9.Step 5: For 5632, digit sum is 5 + 6 + 3 + 2 = 16, which is not divisible by 9.Step 6: Therefore, 5632 is the only number whose digit sum is not a multiple of 9 and is the odd one out.

Verification / Alternative check:
We can quickly cross check by division. 2691, 9900, 2790, and 4509 are all divisible by 9, because their digit sums are 18. For example, 2790 divided by 9 is 310. For 5632, dividing by 9 will not produce an integer. The remainder in such an attempt confirms that 5632 is not a multiple of 9, matching the conclusion from the digit sum method.

Why Other Options Are Wrong:
2691, 9900, 2790, and 4509 all share the same property that their digits sum to 18. This links them through the divisibility rule for 9 and places them in the same category of numbers. Because they match on the key property being tested, none of these four can be considered the odd number out.

Common Pitfalls:
Students sometimes miscalculate digit sums when working quickly, for example adding 2 + 6 + 9 + 1 incorrectly. Another common error is to confuse the rules for divisibility by 3 and 9, although both use digit sums. To avoid these mistakes, practise computing digit sums carefully and remember that for 9, the sum itself must be a multiple of 9. This greatly simplifies many questions involving large numbers.

Final Answer:
5632