A1. Rasta-lover Pair

Codeforces

IDCFA1

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)]，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。对于子任务 1-3，该数为 #cf_span[n]；对于子任务 4-6，该数为 #cf_span[A]。请在一行中输出答案对 #cf_span[109 + 7] 取模的结果。 ## Input 每个子任务包含一个测试用例。输入包含一个数。对于子任务 1-3，该数为 #cf_span[n]；对于子任务 4-6，该数为 #cf_span[A]。 ## Output 请在一行中输出答案对 #cf_span[109 + 7] 取模的结果。 [samples]

**Definitions:** - Let $ n \in \mathbb{Z}^+ $. - Let $ \mathbb{Z}_n^* = \{ p \in \mathbb{Z}^+ \mid 1 \leq p < n, \gcd(p, n) = 1 \} $ be the multiplicative group modulo $ n $. - Let $ \lambda(n) $ denote the Carmichael function: the smallest positive integer $ a $ such that $ p^a \equiv 1 \pmod{n} $ for all $ p \in \mathbb{Z}_n^* $. --- **Subtasks 1–3:** Given $ n $, compute the number of pairs $ (p, q) \in \{1, 2, \dots, n-1\}^2 $ such that there exists $ x \in \mathbb{Z}^+ $ with: $$ p^x \equiv q \pmod{n} $$ That is, $ q $ lies in the cyclic subgroup generated by $ p $ modulo $ n $. Equivalently, count: $$ \left| \left\{ (p, q) \in [1, n-1]^2 \mid q \in \langle p \rangle \subseteq \mathbb{Z}/n\mathbb{Z} \right\} \right| $$ --- **Subtasks 4–6:** Given $ A $, compute: $$ M = \lambda(A) \mod (10^9 + 7) $$ where $ \lambda(A) $ is the Carmichael function of $ A $: the exponent of the multiplicative group modulo $ A $, i.e., the smallest positive integer $ a $ such that for all $ p \in \mathbb{Z}_A^* $, $ p^a \equiv 1 \pmod{A} $. --- **Final Formal Output:** $$ \boxed{ \begin{cases} \displaystyle \sum_{p=1}^{n-1} \left| \langle p \rangle \cap \{1, 2, \dots, n-1\} \right| \mod (10^9 + 7) & \text{if input is } n \text{ (subtasks 1–3)} \\ \displaystyle \lambda(A) \mod (10^9 + 7) & \text{if input is } A \text{ (subtasks 4–6)} \end{cases} } $$

API Response (JSON)

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    "name": "A1. 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
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "CFA1"
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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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      "language": "English"
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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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      "language": "Chinese"
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      "statement_type": "Markdown",
      "content": "**Definitions:**\n\n- Let $ n \\in \\mathbb{Z}^+ $.\n- Let $ \\mathbb{Z}_n^* = \\{ p \\in \\mathbb{Z}^+ \\mid 1 \\leq p < n, \\gcd(p, n) = 1 \\} $ be the multiplicative group modulo $ n $.\n- Let $ \\lambda(n) $ den...",
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