Dice Product 3

AtCoder

IDabc300_e

Time2000ms

Memory256MB

Difficulty—

You have an integer $1$ and a die that shows integers between $1$ and $6$ (inclusive) with equal probability. You repeat the following operation while your integer is strictly less than $N$: * Cast a die. If it shows $x$, multiply your integer by $x$. Find the probability, modulo $998244353$, that your integer ends up being $N$. How to find a probability modulo $998244353$? We can prove that the sought probability is always rational. Additionally, under the constraints of this problem, when that value is represented as $\frac{P}{Q}$ with two coprime integers $P$ and $Q$, we can prove that there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find this $R$. ## Constraints * $2 \leq N \leq 10^{18}$ * $N$ is an integer. ## Input The input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

Output #1

239578645

One of the possible procedures is as follows.

*   Initially, your integer is $1$.
*   You cast a die, and it shows $2$. Your integer becomes $1 \times 2 = 2$.
*   You cast a die, and it shows $4$. Your integer becomes $2 \times 4 = 8$.
*   Now your integer is not less than $6$, so you terminate the procedure.

Your integer ends up being $8$, which is not equal to $N = 6$.
The probability that your integer ends up being $6$ is $\frac{7}{25}$. Since $239578645 \times 25 \equiv 7 \pmod{998244353}$, print $239578645$.

Input #2

Output #2

0

No matter what the die shows, your integer never ends up being $7$.

Input #3

Output #3

183676961

Input #4

979552051200000000

Output #4

812376310

API Response (JSON)

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    "name": "Dice Product 3",
    "description": {
      "content": "You have an integer $1$ and a die that shows integers between $1$ and $6$ (inclusive) with equal probability.   You repeat the following operation while your integer is strictly less than $N$: *   Ca",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc300_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You have an integer $1$ and a die that shows integers between $1$ and $6$ (inclusive) with equal probability.  \nYou repeat the following operation while your integer is strictly less than $N$:\n\n*   Ca...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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