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Home » Page of Cups Reversed? This Simple Swap Unlocks Secret Frequently Ignored! - Dachbleche24

Page of Cups Reversed? This Simple Swap Unlocks Secret Frequently Ignored! - Dachbleche24

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 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:

📌 4k \equiv 1 \pmod{7}

📌 Multiplying both sides by the modular inverse of 4 modulo 7, which is 2 (since $4 \cdot 2 = 8 \equiv 1 \pmod{7}$), we get:

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