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

Introduction:
This question tests your ability to quickly recognize whether a given integer can be a perfect square of a natural number without actually finding its square root. Using quick checks like the last digit and digital root helps you eliminate impossible candidates very efficiently.

Given Data / Assumptions:

• You are given five positive integers: 32761, 42437, 81225, 20164 and 28561.
• Exactly one of these numbers cannot be the square of any natural number.
• You must identify which one is impossible to be a perfect square.

Concept / Approach:
Some simple rules for perfect squares are:
1. The last digit of a perfect square in base 10 can only be 0, 1, 4, 5, 6 or 9.2. If the last digit is 2, 3, 7 or 8, the number cannot be a perfect square.We will first apply the last-digit rule, which is extremely fast.

Step-by-Step Solution:
Step 1: Check the unit (ones) digit of each number.32761 → last digit 1 (possible for a square).42437 → last digit 7 (not possible for a square).81225 → last digit 5 (possible).20164 → last digit 4 (possible).28561 → last digit 1 (possible, in fact 169^2).Step 2: Apply the perfect square last-digit rule.Any number ending in 7 cannot be a perfect square in base 10.Therefore, 42437 cannot be a perfect square.

Verification / Alternative check:
If needed, we can cross check one of the others. For instance, 28561 is 169^2 and 81225 is 285^2, confirming that those are indeed perfect squares. 20164 and 32761 also pass more advanced tests and can correspond to actual squares, whereas 42437 fails immediately due to the last digit rule.

Why Other Options Are Wrong:
The other numbers all have permissible unit digits and in fact correspond to real square numbers or at least are not disqualified by the quick checks. Only 42437 ends in 7, which is impossible for the square of a natural number.

Common Pitfalls:
Candidates sometimes attempt to compute or approximate square roots for each option, which wastes time. Remembering the simple last-digit rule for squares lets you detect impossibilities in seconds.

Final Answer:
The number that cannot be a perfect square is 42437.