The smallest natural number which is a perfect square and which ends in 3 identical digits lies between

### Concept & Logic Perfect squares have strict rules about their ending digits. A perfect square can only end in $0, 1, 4, 5, 6,$ or $9$. Furthermore, if a perfect square ends in multiple identical digits: * It cannot end in $111, 555, 666,$ or $999$ because of modular arithmetic constraints (e.g., an odd square must be a multiple of $8$ plus $1$). * It cannot end in an odd number of zeros (e.g., $000$). * The only possible non-zero repeating identical triplet at the end of a perfect square is $444$. ### Step-by-Step Solution * **Deduction:** The square must end in $444$. Since the number ends in $4$, its square root must end in either $2$ or $8$ (since $2^2=4$ and $8^2=64$). * **Calculation:** Let's test squares ending in $2$ or $8$, starting from smaller numbers, to find one that ends in $444$. * $12^2 = 144$ (Ends in 44, not 444) * $22^2 = 484$ * $28^2 = 784$ * $32^2 = 1024$ * $38^2 = 1444$ We found it! $1444$ is a perfect square ($38^2$) and it ends in three identical digits ($444$). * **Conclusion:** The number $1444$ clearly falls into the range between $1000$ and $2000$. ### Exam Strategy & Shortcut Knowing that $38^2 = 1444$ is a tremendous shortcut for higher-level aptitude exams. $1444$ is the **only** four-digit perfect square ending in three identical digits. Recognizing this instantly points you to the $1000-2000$ bucket. ### Common Pitfall * **Mistake:** Trying to guess numbers ending in $000$ (like $1000$ or $4000$) and thinking they are perfect squares. * **Correction:** A perfect square must have an even number of trailing zeros. Three zeros is mathematically impossible for a perfect square. ### Final Answer **Therefore, the correct answer is 1000 and 2000.**