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Home » The probability is $\boxed\dfrac827$. - Appcentric

The probability is $\boxed\dfrac827$. - Appcentric

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 So only \(n \equiv 2 \pmod{5}\) works.

📌 Now lift to mod 25: Let \(n = 5k + 2\). Plug into \(n^3 \equiv 13 \pmod{25}\):

📌 n^3 = (5k+2)^3 = 125k^3 + 3\cdot25k^2\cdot2 + 3\cdot5k\cdot4 + 8 = 125k^3 + 150k^2 + 60k + 8

📌 60k + 8 \equiv 13 \pmod{25} \Rightarrow 10k + 8 \equiv 13 \pmod{25} \Rightarrow 10k \equiv 5 \pmod{25}

📌 Divide by 5: \(2k \equiv 1 \pmod{5} \Rightarrow k \equiv 3 \pmod{5}\) (

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