In the following question, four three digit numbers are given. Three of these numbers are not divisible by 3, while one number is divisible by 3. Select the number which is divisible by 3 (odd number out).

Introduction / Context:
This is a basic number theory question from aptitude that focuses on divisibility by 3. You are given several three digit numbers and asked to identify which one behaves differently from the others. The key is to use the well known rule for divisibility by 3, which allows you to answer very quickly once you practice it.

Given Data / Assumptions:
- The options are 369, 257, 346, 628, and 517.
- We recall that a number is divisible by 3 if and only if the sum of its digits is divisible by 3.
- Exactly one number in the list is a multiple of 3, while the others are not.
- That single multiple of 3 will be the odd one out.

Concept / Approach:
Instead of performing full division, the most efficient approach is to apply the digit sum test. This test is a standard technique for recognising multiples of 3 and 9. By summing the digits of each number and checking whether the sum is divisible by 3, we can quickly classify each option and identify the special number that does not match the rest.

Step-by-Step Solution:
Step 1: For 369, digit sum is 3 + 6 + 9 = 18, which is divisible by 3.Step 2: For 257, digit sum is 2 + 5 + 7 = 14, which is not divisible by 3.Step 3: For 346, digit sum is 3 + 4 + 6 = 13, which is not divisible by 3.Step 4: For 628, digit sum is 6 + 2 + 8 = 16, which is not divisible by 3.Step 5: For 517, digit sum is 5 + 1 + 7 = 13, which is also not divisible by 3.Step 6: Therefore, 369 is the only number whose digit sum is divisible by 3, so 369 is the only multiple of 3 in the list and is the odd number out.

Verification / Alternative check:
We can also verify directly by dividing 369 by 3. We obtain 369 / 3 = 123 exactly, confirming that it is divisible by 3. For the other numbers, quick mental checks show that no exact integer quotient results when dividing by 3, because their digit sums are not multiples of 3. This alternative check supports our conclusion that 369 is special among the given numbers.

Why Other Options Are Wrong:
257, 346, 628, and 517 all have digit sums that are not multiples of 3. This means they are not divisible by 3 and are consistent with each other. Because they share the same divisibility property, these four numbers form one group and cannot be treated as the odd one out in this question.

Common Pitfalls:
Some learners forget to use the digit sum rule and attempt to perform time consuming long division for each number. Others may miscalculate the sums, especially when working quickly under exam conditions. Training yourself to apply digit sum rules accurately and efficiently will save time across the entire quantitative section of an exam and reduce the chance of silly mistakes.

Final Answer:
369