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doubletree by hilton 5780 major blvd orlando fl 32819 - Parker Core Knowledge

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 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 $\boxed{30}$.

📌 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

📌 So we are looking for the smallest three-digit number $x$ such that $x \equiv 1 \pmod{7}$ and $x \equiv 0 \pmod{60}$.

📌 Let $x = 60k$. Then we solve $60k \equiv 1 \pmod{7}$. Since $60 \equiv 4 \pmod{7}$, we solve:

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