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Home » Leonardo Ninja Turtle: The Hidden Legend You’ve Been Searching For! - Crosslake

Leonardo Ninja Turtle: The Hidden Legend You’ve Been Searching For! - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Solution: The sequence is $11071, 11073, 11075, 11077, 11079$. The sum of an arithmetic sequence is given by $S = \frac{n}{2}(a_1 + a_n)$. Here, $n = 5$, $a_1 = 11071$, and $a_5 = 11079$.

📌 S = \frac{5}{2}(11071 + 11079) = \frac{5}{2}(22150) = 5 \cdot 11075 = 55375

📌 Now compute $55375 \mod 8$. We divide $55375$ by $8$:

📌 \div 8 = 6921 \text{ remainder } 7

📌 Thus, the remainder is $\boxed{7}$.

📌 Question: How many lattice points lie on the hyperbola defined by the equation $x^2 - y^2 = 2025$?

📌 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 $x = \frac{a + b}{2}$ and $y = \frac{b - a}{2}$. For $x$ and $y$ to be integers, $a + b$ and $b - a$ must both be even, so $a$ and $b$ must have the same parity.

📌 We factor $2025 = 3^4 \cdot 5^2$, so the number of positive divisors is $(4+1)(2+1) = 15$. Each divisor pair $(a, b)$ corresponds to a solution, and since $a$ and $b$ must have the same parity, we count such pairs.

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