In the following question, select the odd number pair from the given alternatives: 11 – 121, 13 – 169, 19 – 391 and 21 – 441. In three of these pairs, the second number is the exact square of the first number, while in one pair this square relationship is not satisfied. Which pair is the odd one out?

Introduction / Context:
This question is an odd one out problem involving number pairs and perfect squares. You are given four pairs of numbers and must identify which pair does not follow the same square relationship as the others. Recognising squares and square relationships is very common in quantitative aptitude tests and helps in quickly classifying numerical patterns.

Given Data / Assumptions:

• Number pairs: 11 – 121, 13 – 169, 19 – 391, 21 – 441.
• We suspect that a square relationship might exist because numbers like 121, 169 and 441 are familiar squares.
• We will check whether the second number in each pair equals the square of the first number.
• One pair will fail this test and will be the odd one out.

Concept / Approach:
The basic concept is that of perfect squares:

• n^2 means n multiplied by itself.
• For each pair (a, b), we check whether b = a^2.

If three of the pairs satisfy this exact square relation and one does not, the nonconforming pair is the odd one out. This approach uses straightforward arithmetic and basic knowledge of square numbers.

Step-by-Step Solution:
Step 1: For 11 – 121: Compute 11^2 = 11 * 11 = 121. So the second number is exactly the square of the first. Step 2: For 13 – 169: Compute 13^2 = 13 * 13 = 169. Again, the second number is the square of the first. Step 3: For 21 – 441: Compute 21^2 = 21 * 21 = 441. The second number is exactly equal to 21^2. Step 4: For 19 – 391: Compute 19^2 = 19 * 19 = 361. However, the second number given is 391, which is not equal to 361. Step 5: Because 391 does not equal 19^2, the pair 19 – 391 does not follow the square relationship shared by the other three pairs.

Verification / Alternative check:
You can also check by factorisation: 121 = 11 * 11, therefore a perfect square. 169 = 13 * 13, again a perfect square. 441 = 21 * 21, also a perfect square. 391, however, can be factorised as 17 * 23, which is not of the form n * n. Thus 391 is not a perfect square, confirming that 19 – 391 is structurally different from the other pairs.

Why Other Options Are Wrong:
11 – 121 is not odd because 121 is the perfect square of 11. 13 – 169 is not odd because 169 is the perfect square of 13. 21 – 441 is not odd because 441 is the perfect square of 21. Only 19 – 391 breaks the rule as 391 is not equal to 19^2.

Common Pitfalls:
A frequent mistake is miscalculating the square of 19 or confusing 391 with 361 due to similar digits. Some students might also attempt to relate the numbers using arbitrary multiplication or addition instead of checking the simplest and most obvious square relationship. Always calculate squares carefully and, when in doubt, verify by factorisation.

Final Answer:
The pair that does not follow the square relationship and is therefore the odd one out is 19 – 391.