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📅 May 5, 2026 👤 scraface

May 05, 2026

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📌 36k \equiv 3 \pmod{7} \Rightarrow k \equiv 3 \cdot 36^{-1} \pmod{7}

📌 Since $36 \equiv 1 \pmod{7}$, its inverse modulo 7 is 1. So:

📌 k \equiv 3 \pmod{7} \Rightarrow k = 7m + 3

📌 Then $n = 36k = 36(7m + 3) = 252m + 108$.

📌 We seek the smallest four-digit value of $n$:

📌 252m + 108 \geq 1000 \Rightarrow 252m \geq 892 \Rightarrow m \geq \left\lceil \frac{892}{252} \right\rceil = 4

📌 For $m = 4$:

📌 n = 252 \cdot 4 + 108 = 1008 + 108 = 1116

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