Perfect-square spotting — Which of the following is not a perfect square of any natural number? Use quick tests and known squares.

Introduction / Context:Identifying non-squares quickly is an important test skill. While some options may be recognizable as well-known squares, others require quick checks: digit rules, last-two-digit patterns, and approximating square roots. This question asks you to spot the one number that is not a perfect square.

Given Data / Assumptions:

• Candidates: 34692, 4096, 15129, 15376, 16641.
• We need to determine which is not a square of a natural number.
• Use recognition and short checks rather than long calculations.

Concept / Approach:Recall known squares: 64^2 = 4096; 123^2 = 15129; 124^2 = 15376; 129^2 = 16641. For 34692, check nearby squares by estimating its square root and comparing to exact squares. Also, use quick filters like last-digit feasibility and digital patterns (optional).

Step-by-Step Solution:Confirm known squares: 64^2 = 4096 → square.123^2 = 15129 → square.124^2 = 15376 → square.129^2 = 16641 → square.Evaluate 34692: sqrt(34692) is between 186 (186^2 = 34596) and 187 (187^2 = 34969). Since 34692 lies between these squares, it is not itself a perfect square.

Verification / Alternative check:Compute nearest squares precisely: 186^2 = 34596 and 187^2 = 34969; 34692 is strictly between them, confirming it is not a square.

Why Other Options Are Wrong:

• 4096 = 64^2; 15129 = 123^2; 15376 = 124^2; 16641 = 129^2 — all are exact squares, so they are not the correct choice.

Common Pitfalls:Assuming an unfamiliar number is not a square without checking; rounding square roots too loosely; misremembering common squares like 64^2.

Final Answer:34692