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Home » They’re Obsessed: How Black Dock Wallpaper Adds Stormy Elegance to Any Room! - 500apps

They’re Obsessed: How Black Dock Wallpaper Adds Stormy Elegance to Any Room! - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 \( n \equiv 4 \pmod{5} \Rightarrow n^2 \equiv 16 \equiv 1 \pmod{5} \).

📌 So, \( n^2 \equiv 1 \pmod{5} \) when \( n \equiv 1 \) or \( 4 \pmod{5} \).

📌 Step 3: Combine using Chinese Remainder Theorem

📌 We seek the smallest positive \( n \) such that:

📌 \( n \equiv 1 \text{ or } 2 \pmod{3} \)

📌 \( n \equiv 1 \text{ or } 4 \pmod{5} \)

📌 \( n \equiv 1 \pmod{3}, n \equiv 1 \pmod{5} \Rightarrow n \equiv 1 \pmod{15} \)

📌 \( n \equiv 1 \pmod{3}, n \equiv 4 \pmod{5} \): Solve \( n = 5k + 4 \). Then \( 5k + 4 \equiv 1 \pmod{3} \Rightarrow 2k + 1 \equiv 1 \Rightarrow 2k \equiv 0 \Rightarrow k \equiv 0 \pmod{3} \Rightarrow k = 3m \Rightarrow n = 15m + 4 \). So smallest is \( n = 4 \).

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