In the following number analogy, 13 is related to 167 in a specific way. Using the same rule, 23 will be related to which of the following numbers?

Introduction / Context:
This is a numerical analogy question where the first pair of numbers, 13 and 167, is connected through a specific arithmetic pattern. The task is to discover this pattern and then apply it to the second number, 23, to find the correct related number from the options. Such questions help evaluate the ability to recognise hidden formulas and transformations applied consistently across different pairs of numbers.

Given Data / Assumptions:
- First pair: 13 and 167 form the relation 13 : 167.
- Second pair: 23 : ? must follow the same rule.
- The answer choices are 527, 531, 535, and 525.
- The relationship is expected to be simple enough to be used in a timed aptitude test, typically involving squares, cubes, or small additions and subtractions.

Concept / Approach:
We begin by testing familiar operations involving 13. The square of 13 is 169. The given number 167 is very close to 169, just 2 less. Therefore, one neat pattern is 13^2 minus 2 equals 167. If we assume this is the rule, we should apply the same operation to 23. The square of 23 is 529, and subtracting 2 gives 527. If this result matches an option, the pattern is likely correct. A good check is to see whether this rule feels natural and whether any other option could be produced through a simpler and consistent operation.

Step-by-Step Solution:
Step 1: Compute 13 squared. 13 * 13 = 169. Step 2: Compare 169 with the given partner 167. Notice that 167 = 169 - 2. Step 3: Hypothesise the rule: for each number n, its partner is n^2 - 2. Step 4: Apply this rule to 23. Compute 23 * 23 = 529. Step 5: Subtract 2 to obtain 529 - 2 = 527. Step 6: Check options. The number 527 appears as option A, so the completed pair is 23 : 527.

Verification / Alternative check:
To verify, try alternative simple patterns such as n^2 plus or minus a fixed value other than 2 or combinations like 10n plus a constant. For 13, 10n plus a constant would give 130 plus something, which does not match 167 easily. Other nearby values such as 13^2 minus 1 or 13^2 plus 2 would give 168 or 171, which are not used. The pattern n^2 - 2 fits perfectly and extends cleanly to 23, giving a result that is explicitly listed among the options. This confirms that the pattern is most likely the intended one.

Why Other Options Are Wrong:
Option B, 531, would correspond to 23^2 plus 2, which does not match the behaviour of the first pair. Option C, 535, and option D, 525, likewise do not share the same simple square minus constant relationship when applied to 23. Since analogies require a uniform operation for all pairs, any candidate that cannot be expressed as 23^2 - 2 must be rejected.

Common Pitfalls:
One common pitfall is to focus solely on the difference between 13 and 167 instead of considering the square or cube. Another mistake is to accept any pattern that fits the first pair without testing whether it produces a valid option for the second pair. To avoid this, always try the most common patterns first, such as squares, cubes, and linear expressions, and then check that the same rule produces one and only one reasonable option for the unknown term.

Final Answer:
Using the rule n^2 - 2, the number related to 23 is 527, so the completed analogy is 13 : 167 :: 23 : 527 and option A is correct.