A5. Rasta-lover Pair

Codeforces

IDCFA5

Time15000ms

Memory1024MB

Difficulty—

English · Original

Chinese · Translation

Formal · Original

A pair (p, q) of integer numbers is called Rasta - lover if and only if 1 ≤ p, q < n and there is a positive integer like x such that: px ≡ q (modn) Subtasks 1 - 3: Given n, calculate the number of Rasta - lover pairs modulo 109 + 7. Subtasks 4 - 6: A positive integer p is called n - Rastaly if and only if p < n and there is a positive integer like x such that px ≡ 1 (modn) and p and n are coprimes. For a positive integer n, f(n) is the smallest positive integer a such that for each n - Rastaly number like p, pa ≡ 1 (modn) (this number always exists). If M is equal to for a given A, then you have to calculate M modulo 109 + 7. Subtasks: Each subtask consists of one testcase. Input consists of one number. For subtasks 1-3, it's n and for subtasks 4-6 it's A. Print the answer modulo 109 + 7 in one line. ## Input Each subtask consists of one testcase.Input consists of one number. For subtasks 1-3, it's n and for subtasks 4-6 it's A. ## Output Print the answer modulo 109 + 7 in one line. [samples]

一对整数 #cf_span[(p, q)] 被称为 #cf_span[Rasta - lover]，当且仅当 #cf_span[1 ≤ p, q < n] 且存在一个正整数 #cf_span[x] 使得： #cf_span[px ≡ q] #cf_span[(modn)] 子任务 #cf_span[1 - 3]：给定 #cf_span[n]，计算 #cf_span[Rasta - lover] 对的数量，对 #cf_span[109 + 7] 取模。子任务 #cf_span[4 - 6]：一个正整数 #cf_span[p] 被称为 #cf_span[n - Rastaly]，当且仅当 #cf_span[p < n] 且存在一个正整数 #cf_span[x] 使得 #cf_span[px ≡ 1] #cf_span[(modn)]，并且 #cf_span[p] 与 #cf_span[n] 互质。对于正整数 #cf_span[n]，#cf_span[f(n)] 是最小的正整数 #cf_span[a]，使得对于每个 #cf_span[n - Rastaly] 数 #cf_span[p]，都有 #cf_span[pa ≡ 1] #cf_span[(modn)]（这样的数总是存在）。如果给定 #cf_span[A] 时 #cf_span[M] 等于，则你需要计算 #cf_span[M] 对 #cf_span[109 + 7] 取模的结果。子任务：每个子任务包含一个测试用例。输入包含一个数字。对于子任务 1-3，它是 #cf_span[n]；对于子任务 4-6，它是 #cf_span[A]。 ## Input 每个子任务包含一个测试用例。输入包含一个数字。对于子任务 1-3，它是 #cf_span[n]；对于子任务 4-6，它是 #cf_span[A]。 ## Output 请在一行中输出对 #cf_span[109 + 7] 取模后的答案。 [samples]

**Definitions:** Let $ n \in \mathbb{Z}^+ $. - Let $ R(n) = \left\{ (p, q) \in \mathbb{Z}^2 \mid 1 \leq p, q < n \text{ and } \exists x \in \mathbb{Z}^+ \text{ s.t. } p^x \equiv q \pmod{n} \right\} $. - Let $ S(n) = \left\{ p \in \mathbb{Z}^+ \mid 1 \leq p < n,\ \gcd(p, n) = 1,\ \exists x \in \mathbb{Z}^+ \text{ s.t. } p^x \equiv 1 \pmod{n} \right\} $. - Define $ f(n) = \min \left\{ a \in \mathbb{Z}^+ \mid \forall p \in S(n),\ p^a \equiv 1 \pmod{n} \right\} $. --- **Given:** - For subtasks 1–3: Input is $ n $. Output $ |R(n)| \mod (10^9 + 7) $. - For subtasks 4–6: Input is $ A $. Output $ f(A) \mod (10^9 + 7) $. --- **Formal Output:** $$ \boxed{ \begin{cases} |R(n)| \mod (10^9 + 7) & \text{if input is } n \text{ (subtasks 1–3)} \\ f(A) \mod (10^9 + 7) & \text{if input is } A \text{ (subtasks 4–6)} \end{cases} } $$

API Response (JSON)

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    "name": "A5. Rasta-lover Pair",
    "description": {
      "content": "A pair (p, q) of integer numbers is called Rasta - lover if and only if 1 ≤ p, q < n and there is a positive integer like x such that: px ≡ q (modn) Subtasks 1 - 3:  Given n, calculate the number o",
      "description_type": "Markdown"
    },
    "platform": "Codeforces",
    "limit": {
      "time_limit": 15000,
      "memory_limit": 1048576
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    {
      "statement_type": "Markdown",
      "content": "A pair (p, q) of integer numbers is called Rasta - lover if and only if 1 ≤ p, q < n and there is a positive integer like x such that:\n\npx ≡ q (modn)\n\nSubtasks 1 - 3: \n\nGiven n, calculate the number o...",
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    {
      "statement_type": "Markdown",
      "content": "一对整数 #cf_span[(p, q)] 被称为 #cf_span[Rasta - lover]，当且仅当 #cf_span[1 ≤ p, q < n] 且存在一个正整数 #cf_span[x] 使得：\n\n#cf_span[px ≡ q] #cf_span[(modn)]\n\n子任务 #cf_span[1 - 3]：\n\n给定 #cf_span[n]，计算 #cf_span[Rasta - love...",
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      "statement_type": "Markdown",
      "content": "**Definitions:**\n\nLet $ n \\in \\mathbb{Z}^+ $.\n\n- Let $ R(n) = \\left\\{ (p, q) \\in \\mathbb{Z}^2 \\mid 1 \\leq p, q < n \\text{ and } \\exists x \\in \\mathbb{Z}^+ \\text{ s.t. } p^x \\equiv q \\pmod{n} \\right\\} ...",
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