Among the following numbers, which one cannot be the perfect square of a natural number?

Introduction / Context:
This problem asks us to identify which of the given numbers cannot be expressed as the square of a natural number. Instead of taking square roots on a calculator, we can use estimation and properties of perfect squares to determine which options are valid squares and which one is not. This tests number sense, familiarity with square numbers and basic approximation skills.

Given Data / Assumptions:

- The candidate numbers are 32761, 42437, 81225 and 20164. - All numbers are positive and reasonably close to the squares of integers in the range 100 to 300. - Only one of these cannot be written as n^2 for any natural number n.

Concept / Approach:
We can approximate the square root of each number by comparing it to nearby perfect squares. For example, the square of 180 is 32400, the square of 190 is 36100, and so on. Once we identify a likely integer candidate r such that r^2 is close to the given number, we check r^2 and (r + 1)^2 exactly. If the number matches one of these, it is a perfect square; otherwise it is not. We repeat this process for each option until we find the one that fails to match any integer square.

Step-by-Step Solution:
Step 1: For 32761, note that 180^2 = 32400 and 190^2 = 36100. The number is closer to 180^2, so test 181^2. Step 2: Compute 181^2 = 181 * 181 = 32761, which exactly matches. Hence 32761 is a perfect square. Step 3: For 81225, note that 280^2 = 78400 and 290^2 = 84100. Try 285^2. Step 4: Compute 285^2 = 285 * 285 = 81225, so 81225 is a perfect square. Step 5: For 20164, notice that 140^2 = 19600 and 150^2 = 22500. Try 142^2. Step 6: Compute 142^2 = 142 * 142 = 20164, so 20164 is also a perfect square. Step 7: For 42437, find nearby squares. Compute 206^2 = 42436 and 207^2 = 42849. Step 8: Neither 206^2 nor 207^2 equals 42437. There is no integer whose square is 42437, so this number is not a perfect square.

Verification / Alternative check:
Another way is to observe that perfect squares modulo small bases often follow specific patterns. For example, in base 4 or 8, squares have restricted remainders. However, the direct calculation approach with nearby integer squares is usually faster for numbers in this range. Having checked that three of the numbers match exact squares of 181, 285 and 142, and that 42437 lies strictly between 206^2 and 207^2, we are confident that only 42437 fails to be a square.

Why Other Options Are Wrong:
Option A (32761) equals 181^2, option C (81225) equals 285^2 and option D (20164) equals 142^2. Since each of these is an exact perfect square, they cannot be the answer to the question about which number is not a square.

Common Pitfalls:
Some students rely solely on the last digit or rough approximations and may guess without checking both r^2 and (r + 1)^2. Others may mistakenly believe that because 42437 is close to 42436, it must also be a square. Careful computation of nearby squares is essential to avoid such errors.

Final Answer:
The number that cannot be the square of a natural number is 42437, which corresponds to option B.