In this number analogy, 111 is related to 37. Using the same pattern, which number should complete the analogy 111 : 37 :: 123 : ?

Introduction / Context:
This numerical analogy is based on division by a constant factor. The given pair 111 : 37 suggests that the second number may be a result of dividing the first by some integer. Once this factor is identified, we can apply it to the second number in the analogy to obtain the missing value. Such questions are straightforward but require careful basic arithmetic.

Given Data / Assumptions:
First pair: 111 : 37. Second pair: 123 : ?. The numbers appear to be related by a simple arithmetic operation. Exactly one option should follow the same rule.

Concept / Approach:
We examine the relationship between 111 and 37. Quick checks with multiplication show that 37 * 3 equals 111. This leads to an interpretation where the first number is three times the second, or equivalently, the second number is the first divided by 3. The same operation should then be applied to 123 to maintain the pattern in the analogy.

Step-by-Step Solution:
Step 1: Test whether 111 is divisible by 3. 111 / 3 = 37. Step 2: Interpret the relation. The second number is obtained by dividing the first number by 3. Step 3: Apply the same division to 123. 123 / 3 = 41. Step 4: Check the options. 41 is present as option A and fits the rule exactly.

Verification / Alternative check:
We can cross check by reversing the operation. If 41 is correct, then 41 multiplied by 3 should give back 123. Computing 41 * 3 gives 123, confirming the relationship. Additionally, for 111, 37 * 3 gives 111, so both pairs satisfy the same multiply or divide by 3 rule. No other option maintains this exact pattern for both pairs simultaneously.

Why Other Options Are Wrong:
31 would require the second number to be 31, but 31 * 3 equals 93, not 123, so it does not keep the relation. 21 multiplied by 3 gives 63 and has no connection to 123. 71 multiplied by 3 gives 213, which again does not match 123. 39 does not satisfy the same division rule for 123 and breaks the consistent pattern of dividing by 3.

Common Pitfalls:
One common error is to assume addition or subtraction when multiplication or division is more natural. Another is to forget to check whether the same rule works consistently for both pairs. In this type of question, always identify a clear operation for the first pair and then apply it strictly to the second. If the result does not appear in the options, revisit the reasoning rather than forcing a pattern.

Final Answer:
The number that correctly completes the analogy is 41.