In the number series 270, 261, 279, 252, 289 and 243, which one number is the odd one out based on a simple divisibility pattern?

Introduction / Context:
This is an odd one out question based on divisibility properties inside a number series. All the numbers look fairly close to each other and there is no obvious arithmetic or geometric progression. In such cases, aptitude questions often hide a simple property such as divisibility by a fixed integer, which helps to separate one number from the rest.

Given Data / Assumptions:
The series given in the question is:

• 270
• 261
• 279
• 252
• 289
• 243

We assume that five of these numbers share a common divisibility pattern and that only one number breaks this pattern and therefore is the odd one out.

Concept / Approach:
A very common hidden property is divisibility by 9. A number is divisible by 9 if the sum of its digits is a multiple of 9. We will test every number in the list for divisibility by 9. The one number that is not divisible by 9, while all others are, will be the required odd term in the series.

Step-by-Step Solution:
Step 1: Check 270. The sum of its digits is 2 + 7 + 0 = 9, which is divisible by 9, so 270 is a multiple of 9.Step 2: Check 261. The sum of its digits is 2 + 6 + 1 = 9, so 261 is also divisible by 9.Step 3: Check 279. The sum of its digits is 2 + 7 + 9 = 18, which is divisible by 9, so 279 is a multiple of 9.Step 4: Check 252. The sum of its digits is 2 + 5 + 2 = 9, therefore 252 is divisible by 9.Step 5: Check 243. The sum of its digits is 2 + 4 + 3 = 9, so 243 is also a multiple of 9.Step 6: Check 289. The sum of its digits is 2 + 8 + 9 = 19, which is not divisible by 9, so 289 is not a multiple of 9.

Verification / Alternative check:
We can also express each number as 9 multiplied by another integer. For example, 270 = 9 * 30, 261 = 9 * 29, 252 = 9 * 28 and 243 = 9 * 27. Similarly 279 = 9 * 31. However 289 does not give an integer when divided by 9. This alternative view confirms that 270, 261, 279, 252 and 243 form a neat group of multiples of 9, while 289 does not belong to this group.

Why Other Options Are Wrong:
Numbers 270, 279 and 243 are all exact multiples of 9, just like 261 and 252. Treating any of them as the odd one out would ignore the strong and consistent divisibility pattern and would still leave 289 as the only non multiple of 9 in the list.

Common Pitfalls:
Learners sometimes focus on the order of the terms or try to find a complex formula relating consecutive numbers. In many odd one out problems, it is more effective to test basic number properties such as divisibility by 2, 3, 5 or 9. Here, the divisibility by 9 test quickly reveals the correct answer.

Final Answer:
The only number that is not divisible by 9 in this series is 289.