Matthew Koder Bank of America - Decision Point

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📰 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 📰 Pittsburgh Steelers Vs New York Jets Stats 848904 📰 529 Contribution Limit 5543621 📰 Excel Secrets Revealed Convert Columns To Rows Without Losing A Single Detail 409516 📰 Words Start In D 8590891 📰 Hotel Colorado Bell 8892360 📰 An Sn Sn 1 3N2 5N 3N2 N 2 6N 2 2447547 📰 This Maxoff X X Men Crossover Will Send Shivers Down Your Spine 3285313 📰 The Secret Behind Aspertan Will Change Everything 9386441 📰 The Shocking Truth About Seniat No One Talks Aboutbut You Need To See 1888597 📰 Crush Your Music Needs The Best Mp3 Download App That Works Like Magic 4838503 📰 John Q The Heart Wrenching Story That Will Make You Cry You Wont Look Away 4154987 📰 Jp Morgan Chase 2908181 📰 Rodes Your Dream Teen Bedroom Tested Perfect For Any Teen Style 154924 📰 Play Uno Anytime Anywhere The Ultimate Uno Offline Adventure Emerges 7598062 📰 The Shocking Truth Behind Maha Meaning That Will Blow Your Mind 5449872