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.

February 22, 2026

$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.$

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.
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:
\times 2 = 30
Hence, the total number of lattice points is $\boxed{30}$.

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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