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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Since $ k $ must be a non-negative integer, valid values are $ k = 0, 1, 2, ..., 13 $, which gives 14 values.

📌 Thus, there are $ \boxed{14} $ integers between 1 and 100 that are congruent to 3 modulo 7.

📌 Question: What is the smallest positive integer $ n $ such that $ n^2 \equiv 1 \pmod{12} $?

📌 Solution: We are to find the smallest positive integer $ n $ such that:

📌 n^2 \equiv 1 \pmod{12}

📌 This implies $ n^2 - 1 \equiv 0 \pmod{12} $, or $ (n - 1)(n + 1) \equiv 0 \pmod{12} $.

📌 So $ 12 \mid (n - 1)(n + 1) $.

📌 We test small values of $ n $:

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