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Home » y + \frac1y \geq 2 - Get link 4share

y + \frac1y \geq 2 - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 n^2 = (10k + 4)^2 = 100k^2 + 80k + 16 \equiv 80k + 16 \pmod{100}

📌 We want \( 80k + 16 \equiv 76 \pmod{100} \), so:

📌 80k \equiv 60 \pmod{100}

📌 Divide by 20:

📌 4k \equiv 3 \pmod{5} \implies k \equiv 2 \pmod{5} \quad \text{(since } 4^{-1} \equiv 4 \pmod{5}, \text{ so } k \equiv 4 \cdot 3 = 12 \equiv 2 \pmod{5})

📌 Thus, \( k = 5m + 2 \), so \( n = 10(5m + 2) + 4 = 50m + 24 \). The smallest positive \( n \) is \( n = 24 \). Check: \( 24^2 = 576 \), ends in 76 — valid.

📌 Now consider \( n \equiv 6 \pmod{10} \): \( n = 10k + 6 \).

📌 n^2 = 100k^2 + 120k + 36 \equiv 20k + 36 \pmod{100}

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