In this number analogy question, select the related number from the given alternatives: 72 : 90 :: 156 : ?

Introduction / Context:
This number analogy problem involves factor based reasoning. You are given the pair 72 and 90 and must determine which number relates to 156 in the same manner. Reasoning with products of consecutive integers is a common pattern in aptitude questions and appears frequently in exam settings.

Given Data / Assumptions:
- The first number is 72 and the second number is 90.
- The second starting number is 156 and the options to match are 184, 168, 182, and 176.
- We look for a natural pattern using factors or products that is consistent across both pairs.
- We assume standard integer arithmetic with no hidden tricks such as digit manipulation unless clearly supported.

Concept / Approach:
One useful approach is to express 72 and 90 in terms of products of consecutive integers. Notice that 72 can be written as 8 * 9 and 90 can be written as 9 * 10. This suggests that the rule may relate a product of two consecutive integers to the product of the next two consecutive integers. If that pattern holds, we can apply it to 156 by expressing 156 as a product of consecutive integers and then shifting to the next pair.

Step-by-Step Solution:
Step 1: Factor 72 as 8 * 9, where 8 and 9 are consecutive integers.Step 2: Observe that 90 equals 9 * 10, which is formed by taking the second factor 9 and multiplying it by the next integer 10.Step 3: This indicates the rule: if the first number is n * (n + 1), the second number is (n + 1) * (n + 2).Step 4: Now factor 156. We get 12 * 13, which again are consecutive integers.Step 5: Apply the rule: take 13 and multiply it by the next integer 14, giving 13 * 14 = 182.Step 6: Among the options, 182 appears as choice C, so it is the correct answer.Step 7: The pattern is therefore consistent and factor based.

Verification / Alternative check:
To verify, you can restate the rule for the first pair. If the first number is 8 * 9, then the second is 9 * 10. For the second pair, if the first number is 12 * 13, then the second should be 13 * 14. Both follow exactly the same structure. Check quickly that none of the other options equals 13 * 14. Because 13 * 14 is exactly 182, no other answer can satisfy the rule.

Why Other Options Are Wrong:
- 184: This does not equal 13 * 14 and would require a different, more complicated rule, so it is not consistent with the first pair.
- 168: Although it is near 156, it cannot be written as 13 * 14 and therefore breaks the pattern.
- 176: This also fails to match the factor pattern and appears to be a distractor close in magnitude, not in structure.

Common Pitfalls:
Many students only look at the difference 90 minus 72 equal to 18 and then try to apply a fixed addition to 156. However, 156 plus 18 equals 174, which is not even one of the options. Others may try random multiplication or division without considering factorization into consecutive integers. To avoid these traps, always check whether the given numbers have a meaningful factor structure before trying arbitrary arithmetic.

Final Answer:
Because 72 and 90 can be written as 8 * 9 and 9 * 10, and 156 can be written as 12 * 13, the corresponding second number should be 13 * 14 which equals 182. Thus 182 is the correct answer.