In this number analogy, 545 is related to 196 by a digit sum and square rule; using the same pattern, choose the number that completes the analogy: 545 : 196 :: 173 : ?

Introduction / Context:
This question is a number analogy involving multi digit numbers. The pair 545 and 196 suggests that 196 is calculated from 545 using a rule based on its digits. You must apply the same digit based rule to 173 and select the correct result from the options. These questions develop your skill in spotting relationships between sums of digits and perfect squares.

Given Data / Assumptions:

• The first pair is 545 and 196.
• The second pair is 173 and an unknown number.
• The transformation from 545 to 196 involves using its digits in some arithmetic way.
• The result 196 is a perfect square (14 squared).

Concept / Approach:
Begin by examining the digits of 545. A natural idea is to compute their sum and see how it relates to 196. If 5 + 4 + 5 gives 14 and 14 squared gives 196, we have a clear two step rule: sum the digits, then square the sum. We then apply exactly the same two step process to 173. A correct answer must respect both steps, not just one of them.

Step-by-Step Solution:
Step 1: Identify the digits of 545, which are 5, 4 and 5.Step 2: Compute the sum of these digits: 5 + 4 + 5 = 14.Step 3: Square this sum: 14^2 = 196.Step 4: This matches the second number in the first pair, confirming that the rule is: result equals (sum of digits)^2.Step 5: Apply this rule to 173. Its digits are 1, 7 and 3.Step 6: Sum the digits: 1 + 7 + 3 = 11.Step 7: Square this sum: 11^2 = 121.Step 8: From the options, 121 appears as option B, so that is the correct corresponding number.

Verification / Alternative check:
We can check for competing interpretations, such as multiplying digits or concatenating them, but none of those operations on 545 give 196. The sum followed by squaring is the simplest and most precise rule. Applying any other rule to 173 would fail to give 121 in a consistent way. Also, 121 is a perfect square, which mirrors the structure of 196 as a perfect square of the digit sum, adding confidence to this pattern.

Why Other Options Are Wrong:
Option A, 72, does not arise from squaring any simple sum of digits of 173. Option C, 84, and option D, 41, also cannot be obtained by summing 1, 7 and 3 and applying a square or similar neat transformation. They fail to mirror the exact two step digit sum and square rule that clearly links 545 and 196.

Common Pitfalls:
Some candidates may compute the sum of digits correctly but forget to square the result. Others might incorrectly sum the digits or make a mistake squaring the sum, for example taking 11 squared as 111. Carefully performing both steps and doing a quick mental check of common squares, like 11 squared equals 121, can prevent these errors.

Final Answer:
Using the same digit sum and square rule, 173 corresponds to 121, so option B is correct.