In the following number analogy, 12 is related to 72 in a particular way. Using the same pattern, which number should replace the question mark in 12 : 72 :: 13 : ?

Introduction / Context:
This numerical analogy asks you to identify the arithmetic pattern that transforms 12 into 72 and then apply the same rule to 13. Reasoning questions of this type often rely on simple multiplication by a constant factor and are designed to test recognition of straightforward numeric relationships under time pressure.

Given Data / Assumptions:
First pair: 12 : 72. Second pair: 13 : ?. All numbers are positive integers. The same arithmetic rule must connect both pairs.

Concept / Approach:
We start by checking whether 72 is a simple multiple of 12, because multiplication by a whole number is the most common pattern. If we find such a multiplier, we then test it on 13. The idea is to look for a clean and exact relationship rather than a complicated expression that would be unlikely in a typical aptitude test.

Step-by-Step Solution:
Step 1: Check if 72 is divisible by 12. 72 / 12 = 6. So 72 can be written as 12 * 6. Step 2: Interpret the pattern. The second number is 6 times the first number. Step 3: Apply the same rule to the second pair. We take 13 and multiply it by 6. 13 * 6 = 78. Step 4: Match the result with the answer options. 78 is present as one of the options, so it is the correct answer.

Verification / Alternative check:
We can restate the pattern as: second number = first number * 6. For the first pair, 12 * 6 = 72, which is correct. For the second pair, 13 * 6 = 78, which matches our chosen value. No other option satisfies this simple, consistent multiplication rule. Any number other than 78 would either require a different multiplier or an additional operation, which would break the clear pattern shown in the first pair.

Why Other Options Are Wrong:
31 is not 6 times 13 and appears unrelated to a simple multiplicative pattern. 84 is 7 times 12, but the first pair clearly uses 6 as the multiplier, not 7. 100 cannot be expressed as 13 multiplied by an integer that also fits the first pair. 65 is 5 times 13, which uses a different multiplier and therefore does not respect the established rule.

Common Pitfalls:
Some candidates may overcomplicate the problem by trying to involve addition of squares, cubes, or digit manipulations, but the simplest explanation is often best. Another common error is to miscalculate 12 * 6 or 13 * 6 under time pressure. Always verify quick mental multiplication by reversing it through division, and confirm that the same exact rule works for all given pairs before selecting your final answer.

Final Answer:
The number that correctly completes the analogy is 78.