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        "content": "To find the least odd prime factor of \\(2019^8 + 1\\), we start by noting that if a prime \\( p \\) divides \\(2019^8 + 1\\), then \\(2019^8 \\equiv -1 \\pmod{p}\\). This implies \\(2019^{16} \\equiv 1 \\pmod{p}\\), so the order of 2019 modulo \\( p \\) is 16. By Fermat's Little Theorem, the order must divide \\( p - 1 \\), hence \\( p \\equiv 1 \\pmod{16} \\).\n\nThe primes congruent to 1 modulo 16 are 17, 97, 113, 193, etc. We check the smallest primes:\n\n1. **Prime 17**: Compute \\(2019 \\mod 17\\):\n   - \\(2019 \\equiv 13 \\pmod{17}\\)\n   - Calculate \\(13^8 \\mod 17\\):\n     - \\(13^2 \\equiv 16 \\pmod{17}\\)\n     - \\(13^4 \\equiv 16^2 \\equiv 1 \\pmod{17}\\)\n     - \\(13^8 \\equiv 1^2 \\equiv 1 \\pmod{17}\\)\n   - Thus, \\(2019^8 + 1 \\equiv 1 + 1 = 2 \\pmod{17}\\), so 17 is not a factor.\n\n2. **Prime 97**: Compute \\(2019 \\mod 97\\):\n   - \\(2019 \\equiv 79 \\pmod{97}\\)\n   - Calculate \\(79^8 \\mod 97\\):\n     - \\(79^2 \\equiv 33 \\pmod{97}\\)\n     - \\(79^4 \\equiv 33^2 \\equiv 22 \\pmod{97}\\)\n     - \\(79^8 \\equiv 22^2 \\equiv 96 \\pmod{97}\\)\n   - Thus, \\(2019^8 + 1 \\equiv 96 + 1 = 97 \\equiv 0 \\pmod{97}\\), so 97 is a factor.\n\nSince 97 is the smallest prime congruent to 1 modulo 16 that divides \\(2019^8 + 1\\), the least odd prime factor is \\(\\boxed{97}\\).",
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        "content": "{\n  \"binary_pass_fail\": true,\n  \"additional_feedback\": \"The submission accurately determined the least odd prime factor of 2019^8 + 1 as 97, following a thorough mathematical approach.\"\n}",
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