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Home » P(X = 2) = \frac\binom52 \binom72\binom124 - Ready Digital AB

P(X = 2) = \frac\binom52 \binom72\binom124 - Ready Digital AB

📅 March 7, 2026 👤 scraface

Mar 07, 2026

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📌 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.

📌 Since $2025$ is odd, all its divisors are odd, so all pairs $(a, b)$ have the same parity. Thus, each of the 15 positive divisor pairs gives a solution. Including negative divisors (since $(-a)(-b) = 2025$), we double this count:

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