A5. Rasta-lover Pair

Codeforces

IDCF10070A5

Time15000ms

Memory1024MB

Difficulty—

English · Original

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]

**Definitions** Let $ n \in \mathbb{Z}^+ $. Let $ R(n) = \{ (p, q) \in \mathbb{Z}^2 \mid 1 \leq p, q < n \land \exists x \in \mathbb{Z}^+ \text{ s.t. } p^x \equiv q \pmod{n} \} $ be the set of Rasta-lover pairs. Let $ S(n) = \{ p \in \mathbb{Z}^+ \mid 1 \leq p < n \land \gcd(p, n) = 1 \land \exists x \in \mathbb{Z}^+ \text{ s.t. } p^x \equiv 1 \pmod{n} \} $ be the set of $ n $-Rastaly numbers. Let $ f(n) = \min \{ a \in \mathbb{Z}^+ \mid \forall p \in S(n),\ p^a \equiv 1 \pmod{n} \} $. **Constraints** - For subtasks 1–3: Input is $ n $, compute $ |R(n)| \mod (10^9 + 7) $. - For subtasks 4–6: Input is $ A $, compute $ f(A) \mod (10^9 + 7) $. **Objective** Given input $ X $: - If $ X $ corresponds to subtasks 1–3: output $ |R(n)| \mod (10^9 + 7) $. - If $ X $ corresponds to subtasks 4–6: output $ f(A) \mod (10^9 + 7) $.

API Response (JSON)

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  "problem": {
    "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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    "difficulty": "None",
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    "sign": "CF10070A5"
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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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      "content": "**Definitions**  \nLet $ n \\in \\mathbb{Z}^+ $.  \nLet $ R(n) = \\{ (p, q) \\in \\mathbb{Z}^2 \\mid 1 \\leq p, q < n \\land \\exists x \\in \\mathbb{Z}^+ \\text{ s.t. } p^x \\equiv q \\pmod{n} \\} $ be the set of Ras...",
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