Choose the odd number from the following alternatives based on divisibility by 3.

Introduction / Context:
This question evaluates your familiarity with simple divisibility rules, particularly divisibility by 3. Recognizing such patterns quickly is extremely helpful in number series, simplification, and odd one out tasks in competitive exams.

Given Data / Assumptions:

• The numbers presented are: 165, 792, 852, and 671.
• You must determine which number does not follow the same divisibility pattern as the others.
• Use the standard rule that a number is divisible by 3 if the sum of its digits is divisible by 3.

Concept / Approach:
The divisibility rule for 3 is simple and fast. You add the digits of a number. If the resulting sum is a multiple of 3, then the original number is divisible by 3. By applying this rule to each option, you can quickly group numbers into those divisible by 3 and those that are not, then choose the odd one out.

Step-by-Step Solution:
Step 1: For 165, add the digits: 1 + 6 + 5 = 12. Since 12 is divisible by 3, 165 is divisible by 3. Step 2: For 792, add the digits: 7 + 9 + 2 = 18. Since 18 is divisible by 3, 792 is divisible by 3. Step 3: For 852, add the digits: 8 + 5 + 2 = 15. Since 15 is divisible by 3, 852 is divisible by 3. Step 4: For 671, add the digits: 6 + 7 + 1 = 14. Since 14 is not divisible by 3, 671 is not divisible by 3. Step 5: Three numbers are divisible by 3, and one number is not, so the one that is not divisible by 3 is the odd one out.

Verification / Alternative check:
You can verify the conclusion by performing quick mental division. For example, 165 divided by 3 gives 55, 792 divided by 3 gives 264, and 852 divided by 3 gives 284, all of which are exact integers. For 671, dividing by 3 results in a non integer value. This cross check confirms the result obtained from the digit sum method.

Why Other Options Are Wrong:

• 165: Divisible by 3 based on the digit sum of 12, so it matches the main pattern.
• 792: Clearly divisible by 3 as the digit sum 18 is a multiple of 3.
• 852: Also divisible by 3 since the digit sum 15 is a multiple of 3.

Common Pitfalls:
Students sometimes attempt long division for each number, which is time consuming and can lead to mistakes under exam pressure. Another mistake is to assume that any number ending in 5 or 0 is special only with respect to 5 or 10, but here the relevant property is divisibility by 3. Using the digit sum rule makes the solution faster and more reliable.

Final Answer:
671