In this number pair reasoning question, each option shows a pair in the form a – b: 10 – 121, 12 – 169, 14 – 225 and 16 – 279. Select the odd pair that does not follow the same rule as the others.

Introduction / Context:
This question tests your ability to detect a numerical relationship between two numbers in a pair. The pairs given are 10 – 121, 12 – 169, 14 – 225 and 16 – 279. In three of these pairs, the second number is the square of the successor of the first number. One pair does not follow this relationship and must be identified as the odd one out.

Given Data / Assumptions:

• Number pairs: 10 – 121, 12 – 169, 14 – 225, 16 – 279.
• We consider the relationship between each first number n and the second number.
• In valid pairs, the second number should equal (n + 1)^2.
• Exactly one pair does not satisfy this simple square of successor rule.

Concept / Approach:
The pattern suggested by 10 – 121 is that 121 equals 11^2, which is (10 + 1)^2. We can test this idea on the other pairs. If three pairs fit the rule second = (first + 1)^2 and one pair does not, that mismatching pair is the odd one out. This approach uses only basic knowledge of squares of small integers.

Step-by-Step Solution:
Step 1: Check 10 – 121.First number n = 10. Successor is n + 1 = 11, and 11^2 = 121. So this pair matches the rule second = (n + 1)^2.Step 2: Check 12 – 169.First number n = 12. Successor is 13, and 13^2 = 169. So this pair also follows the rule.Step 3: Check 14 – 225.First number n = 14. Successor is 15, and 15^2 = 225. Again, the second number equals (n + 1)^2, so this pair is consistent.Step 4: Check 16 – 279.First number n = 16. Successor is 17. Here 17^2 = 289, but the second number given is 279, which is not equal to 289. So this pair does not follow the established pattern.Step 5: Identify the odd pair.The first three pairs satisfy the rule second = (first + 1)^2, while 16 – 279 does not. Therefore 16 – 279 is the odd one out.

Verification / Alternative check:
Another way to check is to take the square root of the second number and then subtract 1. For the pair 10 – 121, sqrt(121) = 11 and 11 - 1 = 10. For 12 – 169, sqrt(169) = 13 and 13 - 1 = 12. For 14 – 225, sqrt(225) = 15 and 15 - 1 = 14. For 16 – 279, the square root of 279 is not an integer and certainly not equal to 17, so the pattern fails. This confirms independently that the last pair is inconsistent with the rule.

Why Other Options Are Wrong:
10 – 121: Both numbers satisfy the rule second = (first + 1)^2, so this pair is correct.
12 – 169: Also matches the rule and belongs to the main pattern.
14 – 225: Follows the same relationship, so it cannot be the odd pair.

Common Pitfalls:
Some candidates may try to relate the second number directly to the square of the first number and get confused when the match does not work. Always remember to check for the square of the successor or predecessor as well, because exam setters often use such small variations to increase difficulty slightly. Once you train yourself to test (n + 1)^2 and (n - 1)^2 in addition to n^2, these odd one out questions become much easier.

Final Answer:
The odd pair is 16 – 279, because in this pair the second number is not equal to the square of the successor of the first number, unlike the other three pairs.