Solution: The area of the city is $2025$ km², and we are to find how many integer side lengths $s$ correspond to square grid sizes such that $s^2$ divides $2025$. This requires finding the number of positive divisors of $2025$, since each such divisor $s^2$ implies $s$ is an integer divisor of $\sqrt{2025} = 45$.

["Title: How Many Integer Side Lengths Fit in a 2025 km² City Grid? (Find Integer S Such That $s^2 \mid 2025$)", "Urban planning and spatial analysis often involve optimizing grid layouts within defined areas. Suppose a city spans a total area of $2025$ km². Planners want to explore all possible square grid sizes—where each side has integer length $s$—such that the total area occupied by the grid, $s^2$, divides evenly into the city’s area. This leads to the mathematical solution: determining how many positive integers $s$ satisfy $s^2 \mid 2025$.", "### Understanding the Problem", "We are not looking for just any integer $s$, but specifically integer values of $s$ where $s^2$ divides $2025$. Since $s^2$ must divide $2025$, and $s$ is an integer, this problem reduces to finding the number of positive integer divisors $s$ such that $s^2$ is a divisor of $2025$.", "But note: $s^2 \mid 2025$ implies that $s$ must be a divisor of $\sqrt{2025}$, provided $\sqrt{2025}$ is an integer. Let’s verify this step carefully.", "### Factorization of 2025", "Begin by factoring $2025$:", "[\n2025 = 25 \ imes 81 = (5^2)(3^4) = 3^4 \ imes 5^2\n]", "So the prime factorization is:\n[\n2025 = 3^4 \ imes 5^2\n]", "We seek all positive integers $s$ such that $s^2 \mid 3^4 \ imes 5^2$.", "### Expressing $s$ in Exponential Form", "Let $s = 3^a \ imes 5^b$, where $a, b$ are non-negative integers. Then:", "[\ns^2 = 3^{2a} \ imes 5^{2b}\n]", "For $s^2 \mid 3^4 \ imes 5^2$, we require:\n[\n2a \leq 4 \quad \ ext{and} \quad 2b \leq 2\n]", "Solving these inequalities:\n- $2a \leq 4 \Rightarrow a \leq 2$, so $a = 0, 1, 2$ (3 choices)\n- $2b \leq 2 \Rightarrow b \leq 1$, so $b = 0, 1$ (2 choices)", "### Counting Valid Integer Side Lengths", "Each valid pair $(a, b)$ gives a unique integer $s = 3^a \ imes 5^b$, and since $a$ has 3 options and $b$ has 2, the total number of such $s$ is:", "[\n3 \ imes 2 = 6\n]", "Thus, there are 6 integer side lengths $s$ such that $s^2$ divides $2025$, corresponding to perfect square divisors of $2025$. These $s$ values represent valid square grid sizes (in km) that evenly fit within the 2025 km² city area.", "### Final Answer", "The number of integer side lengths $s$ where $s^2$ divides $2025$ is $6$.", "Keywords: integer side lengths, square grid size, divisor count, $s^2 \mid 2025$, city area grid, factorization, perfect square divisor, urban planning math, number theory application."]