Select the related number from the given alternatives: 21 : 3 :: 574 : ?

Introduction / Context:
This numerical analogy problem tests your ability to recognize a hidden arithmetic operation connecting two numbers. The pair 21 : 3 provides the pattern, and you must apply it to 574 to determine the answer. Often such questions involve division by a constant or using a factor such as a well known number that works for both pairs.

Given Data / Assumptions:
- The first pair is 21 and 3, the second pair is 574 and an unknown number.
- There should be a single, simple arithmetic rule connecting 21 to 3 that can also connect 574 to one of the options.
- We assume standard operations such as division, multiplication, and sums of digits before checking complicated patterns.
- Only one of 23, 82, 97, and 113 fits the same rule.

Concept / Approach:
Start by considering simple relationships between 21 and 3. We quickly notice that 21 divided by 7 equals 3. Thus, the rule may be "divide by 7." If this rule is correct, then applying it to 574 should produce one of the options. This is much simpler and more natural than building an elaborate combination of operations only to force one option.

Step-by-Step Solution:
Step 1: Analyze the first pair 21 and 3. Compute 21 / 7 = 3. Step 2: Hypothesize that the transformation rule is "divide the first number by 7 to get the second." Step 3: Apply this rule to 574. Compute 574 / 7. Step 4: Perform the division. 7 times 80 is 560, and 574 - 560 is 14. 7 times 2 is 14, so 574 / 7 = 82. Step 5: Check if 82 appears in the options. It does, as option B. Step 6: Therefore, 574 is related to 82 in the same way that 21 is related to 3.

Verification / Alternative check:
Try other simple rules such as 21 minus 18 equals 3, but that would require subtracting 18 from 574 and gives 556, which is not among the options. Using 21 divided by the sum of digits (2 + 1 = 3) gives 7, not 3. So division by 7 is the cleanest and most natural rule. When applied to 574, it yields 82, which fits exactly. This confirms the correctness of our interpretation.

Why Other Options Are Wrong:
Option A "23" would require a different divisor, and no simple arithmetic rule from 21 to 3 would also make 574 go to 23 in a consistent way.
Option C "97" and option D "113" also fail to be 574 divided by any simple factor that connects back nicely to 21 and 3. They would require different and more complex rules that do not match the first pair.

Common Pitfalls:
A typical mistake is to overcomplicate the pattern, trying to combine multiple operations when a single clean operation, such as division by a common factor, is available. Another error is to use the sum or product of digits without confirming that the same approach works in both pairs. Always test simple relationships first and check that they produce one of the answer choices directly.

Final Answer:
The correct related number is 82.