= 45^2 = (3^2 \cdot 5)^2 = 3^4 \cdot 5^2

February 22, 2026

$= 45^2 = (3^2 \cdot 5)^2 = 3^4 \cdot 5^2$

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Question: A geographer is analyzing population clusters using a grid where each cell represents a 1 km² area. If a city covers a region that is a perfect square of area $2025$ km², how many different square grid sizes (with integer side lengths) can evenly divide the area?
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 $\sqrt{2025} = 45$.
First, factor $2025$:
The number of positive divisors of $2025$ is given by multiplying one more than each exponent in its prime factorization:
Each divisor corresponds to a perfect square divisor $s^2$, so the number of such $s$ (i.e., integer side lengths of square grids dividing the area) is equal to the number of positive divisors of $2025$, which is $15$. However, each divisor $d$ of $2025$ corresponds to $s = \sqrt{d}$, but only those $d$ that are perfect squares yield integer $s$. Since $2025 = 45^2$, the number of square divisors equals the number of perfect square divisors.
To count perfect square divisors: for a divisor to be a perfect square, all exponents in its prime factorization must be even.

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