Sequence of Multiples

AtCoder

IDarc135_e

Time6000ms

Memory256MB

Difficulty—

You are given integers $N$ and $X$. Assume that an integer sequence $A = (A_1, \ldots, A_N)$ satisfies all of the conditions below. * $A_1 = X$. * For every $i$ ($1\leq i\leq N$), $A_i$ is a multiple of $i$. * $A$ is strictly increasing. In other words, $A_1 < \cdots < A_N$ holds. Find the minimum possible value of $\sum_{i=1}^N A_i$, modulo $998244353$. There are $T$ test cases, each of which should be solved. ## Constraints * $1\leq T\leq 10$ * $1\leq N \leq 10^{18}$ * $1\leq X \leq 10^{18}$ ## Input Input is given from Standard Input from the following format: $T$ $\text{case}_1$ $\vdots$ $\text{case}_T$ Each case is in the following format: $N$ $X$ [samples]

Samples

Input #1

5
5 100
1 10
10 1
1000000000000000000 1
100 100

Output #1

525
10
55
75433847
61074

Here is a sequence $A$ that minimizes $\sum_{i=1}^N A_i$ for each of the first three test cases.

*   The first test case: $A = (100, 102, 105, 108, 110)$.
*   The second test case: $A = (10)$.
*   The third test case: $A = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)$.

API Response (JSON)

{
  "problem": {
    "name": "Sequence of Multiples",
    "description": {
      "content": "You are given integers $N$ and $X$. Assume that an integer sequence $A = (A_1, \\ldots, A_N)$ satisfies all of the conditions below. *   $A_1 = X$. *   For every $i$ ($1\\leq i\\leq N$), $A_i$ is a mult",
      "description_type": "Markdown"
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    "platform": "AtCoder",
    "limit": {
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      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc135_e"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given integers $N$ and $X$. Assume that an integer sequence $A = (A_1, \\ldots, A_N)$ satisfies all of the conditions below.\n\n*   $A_1 = X$.\n*   For every $i$ ($1\\leq i\\leq N$), $A_i$ is a mult...",
      "is_translate": false,
      "language": "English"
    }
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}

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