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Home » Combined rate: \(1 + \frac57 = \frac127\% \) per hour - SUpost

Combined rate: \(1 + \frac57 = \frac127\% \) per hour - SUpost

📅 July 6, 2026 👤 scraface

Jul 06, 2026

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📌 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

📌 Try $m = 0$: $x = 120$ (valid, three-digit). Check $120 \mod 7 = 1$?

📌 \div 7 = 17 \text{ remainder } 1 \Rightarrow \text{Yes}

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