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Home » The general solution can be parameterized in terms of $y$: - AssociationVoting

The general solution can be parameterized in terms of $y$: - AssociationVoting

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 \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:

📌 k \equiv 2 \pmod{7}

📌 So $k = 7m + 2$. Then $x = 60k = 60(7m + 2) = 420m + 120$. We seek the smallest three-digit such $x$:

📌 x = 420m + 120 \geq 100 \Rightarrow m \geq 0

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