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

February 22, 2026

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

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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:
\times 2 = 30
Question: What is the smallest three-digit number that is divisible by both $12$ and $15$ and leaves a remainder of $1$ when divided by $7$?
Solution: A number divisible by both $12$ and $15$ must be divisible by their least common multiple:
\text{lcm}(12, 15) = 60

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