{
  "problem": "Find $24^{-1} \\pmod{11^2}$. That is, find the residue $b$ for which $24b \\equiv 1\\pmod{11^2}$.\n\nExpress your answer as an integer from $0$ to $11^2-1$, inclusive.",
  "level": "Level 5",
  "type": "Number Theory",
  "solution": "Since $5 \\times 24 = 120 = 121 - 1$, it follows that $-5 \\times 24 \\equiv 1 \\pmod{121}$. Adding 121 to $-5$ to make it positive, we find $(-5 + 121) \\times 24 \\equiv 116 \\times 24 \\equiv 1 \\pmod{121}$, so it follows that the modular inverse of $24$ is $\\boxed{116}$ when taken modulo $121$.",
  "correct_ans": "116",
  "problem_id": "5",
  "response_with_ans": "To find the modular inverse of 24 modulo 11^2, we can use the Extended Euclidean Algorithm. Here is a Python function to compute the modular inverse using this algorithm:\n\n```python\ndef mod_inverse(a, m):\n    g, x, _ = extended_gcd(a, m)\n    if g != 1:\n        raise Exception(f\"{a} and {m} are not coprime.\")\n    return x % m\n\ndef extended_gcd(a, b):\n    if a == 0:\n        return b, 0, 1\n    else:\n        g, x, y = extended_gcd(b % a, a)\n        return g, y - (b // a) * x, x\n```\n\nLet's use the above function to find the modular inverse of 24 modulo 11^2:\n\n```python\na = 24\nm = 11**2\nmod_inverse(a, m)\n```\n\nI will execute the above code to find the modular inverse of 24 modulo 11^2.",
  "round": 0,
  "messages": [
    {
      "content": "Find $24^{-1} \\pmod{11^2}$. That is, find the residue $b$ for which $24b \\equiv 1\\pmod{11^2}$.\n\nExpress your answer as an integer from $0$ to $11^2-1$, inclusive.",
      "role": "user"
    },
    {
      "content": "To find the modular inverse of 24 modulo 11^2, we can use the Extended Euclidean Algorithm. Here is a Python function to compute the modular inverse using this algorithm:\n\n```python\ndef mod_inverse(a, m):\n    g, x, _ = extended_gcd(a, m)\n    if g != 1:\n        raise Exception(f\"{a} and {m} are not coprime.\")\n    return x % m\n\ndef extended_gcd(a, b):\n    if a == 0:\n        return b, 0, 1\n    else:\n        g, x, y = extended_gcd(b % a, a)\n        return g, y - (b // a) * x, x\n```\n\nLet's use the above function to find the modular inverse of 24 modulo 11^2:\n\n```python\na = 24\nm = 11**2\nmod_inverse(a, m)\n```\n\nI will execute the above code to find the modular inverse of 24 modulo 11^2.",
      "role": "assistant"
    }
  ],
  "time": 13.481226921081543,
  "check_result": "let me verify the modular inverse of 24 modulo 11^2.\n\nafter executing the provided python code:\n\n```python\na = 24\nm = 11**2\nmod_inverse(a, m)\n```\n\nthe answer is found to be: 116.\n\nthe answer is correct.",
  "is_correct": true,
  "trial": -1
}