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Home » $ S(1,1) = 1 $ - Rice University

$ S(1,1) = 1 $ - Rice University

📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 \text{LCM}(12, 18) = 2^2 \cdot 3^2 = 4 \cdot 9 = 36

📌 Thus, the smallest positive angle that returns the robot to its starting orientation is $ \boxed{36^\circ} $.

📌 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 $2025$:

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

📌 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.

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