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bolongo bay beach resort st thomas all inclusive - inBeat

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 The $ y $-intercept occurs when $ x = 0 $:

📌 y = 3(0) + 1 = 1

📌 Thus, the $ y $-intercept point is $ oxed{(0, 1)} $.

📌 Question: If $ a + b = 7 $ and $ ab = 10 $, what is the value of $ a^3 + b^3 $?

📌 Substitute the given values $ a + b = 7 $, $ ab = 10 $:

📌 a^3 + b^3 = 7^3 - 3(10)(7) = 343 - 210 = 133

📌 Thus, the value is $ oxed{133} $.Question: How many lattice points lie on the hyperbola $ x^2 - y^2 = 2025 $?

📌 Solution: The equation $ x^2 - y^2 = 2025 $ factors as $ (x - y)(x + y) = 2025 $. Since $ x $ and $ y $ are integers, both $ x - y $ and $ x + y $ must be integers. Let $ a = x - y $ and $ b = x + y $, so $ ab = 2025 $. Then $ x = rac{a + b}{2} $ and $ y = rac{b - a}{2} $. For $ x $ and $ y $ to be integers, $ a + b $ and $ b - a $ must both be even, meaning $ a $ and $ b $ must have the same parity. Since $ 2025 = 3^4 \cdot 5^2 $, it has $ (4+1)(2+1) = 15 $ positive divisors. Each pair $ (a, b) $ such that $ ab = 2025 $ gives a solution, but only those with $ a $ and $ b $ of the same parity are valid. Since 2025 is odd, all its divisors are odd, so $ a $ and $ b $ are both odd, ensuring $ x $ and $ y $ are integers. Each positive divisor pair $ (a, b) $ with $ a \leq b $ gives a unique solution, and since 2025 is a perfect square, there is one square root pair. There are 15 positive divisors, so 15 such factorizations, but only those with $ a \leq b $ are distinct under sign and order. Considering both positive and negative factor pairs, each valid $ (a,b) $ with $ a

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