In the following question, select the odd number from the given alternatives: 133, 253, 231 and 209. Three of these numbers are divisible by 11, while one number is not divisible by 11. Based on this divisibility property, which number is the odd one out?

Introduction / Context:
This question checks your understanding of divisibility rules and prime factorisation. You are given four numbers and asked to select the one that does not share a particular divisibility property with the others. Here, three of the numbers are divisible by 11, while one number is not. Identifying such patterns is common in odd one out questions in quantitative aptitude exams.

Given Data / Assumptions:

• Given numbers: 133, 253, 231, 209.
• We will use divisibility checks and factorisation.
• We suspect a common factor, specifically 11, in three of the numbers.
• One number will not be divisible by 11 and thus be the odd one out.

Concept / Approach:
The approach is to check each number for divisibility by 11:

• Use either the difference-of-sums rule for 11 or straightforward factorisation.
• If 11 divides three numbers but not the fourth, the undivided number is the odd one out.

You can also factor each number directly to confirm which ones contain 11 as a factor.

Step-by-Step Solution:
Step 1: Check 253. Try dividing by 11: 11 * 23 = 253, so 253 is divisible by 11. Step 2: Check 231. Try dividing by 11: 11 * 21 = 231, so 231 is divisible by 11. Step 3: Check 209. Try dividing by 11: 11 * 19 = 209, so 209 is also divisible by 11. Step 4: Check 133. Try dividing by 11: 11 * 12 = 132, and 11 * 13 = 143, so 133 lies between these and is not a multiple of 11. Step 5: Alternatively, factor 133: Try 7: 7 * 19 = 133, so 133 = 7 * 19 and contains no factor 11. Step 6: Therefore, 253, 231 and 209 are all divisible by 11, while 133 is not. Hence 133 is the odd one out.

Verification / Alternative check:
You can also use the divisibility rule for 11, which says: If the difference between the sum of the digits in odd places and the sum of the digits in even places is a multiple of 11 (including 0), then the number is divisible by 11. Apply this: For 253: (2 + 3) - 5 = 5 - 5 = 0 → divisible by 11. For 231: (2 + 1) - 3 = 3 - 3 = 0 → divisible by 11. For 209: (2 + 9) - 0 = 11 - 0 = 11 → divisible by 11. For 133: (1 + 3) - 3 = 4 - 3 = 1, which is not a multiple of 11 → not divisible by 11. This confirms that 133 is the only number not divisible by 11.

Why Other Options Are Wrong:
253 is not odd because it clearly equals 11 * 23 and is divisible by 11. 231 is not odd because it equals 11 * 21 and is divisible by 11. 209 is not odd because it equals 11 * 19 and is divisible by 11. 133 is the only number that lacks 11 as a factor, making it the odd one out.

Common Pitfalls:
Students sometimes rely on rough mental estimation instead of precise multiplication and may incorrectly assume 133 is 11 * 11 or 11 * 12. Another pitfall is to look for patterns in the sum of digits or parity (odd or even) instead of focusing on the intended divisibility property. Using the proper divisibility rule for 11 or exact factorisation prevents these errors.

Final Answer:
The number that is not divisible by 11 and is therefore the odd one out is 133.