Floor, Ceil - Decomposition

AtCoder

IDarc135_a

Time2000ms

Memory256MB

Difficulty—

There is an integer $X$ written on a blackboard. You can do the operation below any number of times (possibly zero). * Choose an integer $x$ written on the blackboard. * Erase one $x$ from the blackboard and write two new integers $\lfloor \frac{x}{2}\rfloor$ and $\lceil \frac{x}{2}\rceil$. Find the maximum possible product of the integers on the blackboard after your operations, modulo $998244353$. What are $\lfloor \frac{x}{2}\rfloor$ and $\lceil \frac{x}{2}\rceil$?For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not greater than $x$, and $\lceil x\rceil$ denotes the smallest integer not less than $x$. For example, the following holds. * For $x = 2$, we have $\lfloor \frac{x}{2}\rfloor = 1$ and $\lceil \frac{x}{2}\rceil = 1$. * For $x = 3$, we have $\lfloor \frac{x}{2}\rfloor = 1$, $\lceil \frac{x}{2}\rceil = 2$. ## Constraints * $1\leq X \leq 10^{18}$ ## Input Input is given from Standard Input from the following format: $X$ [samples]

Samples

Input #1

Output #1

192

Here is a sequence of operations that makes the product of the integers on the blackboard $192$.

*   The initial state of the blackboard is $(15)$.
*   An operation with $x = 15$ changes it to $(7, 8)$.
*   An operation with $x = 7$ changes it to $(8, 3, 4)$.
*   An operation with $x = 4$ changes it to $(8, 3, 2, 2)$.
*   An operation with $x = 8$ changes it to $(3, 2, 2, 4, 4)$.

Here, the product of the integers on the blackboard is $3\times 2\times 2\times 4\times 4 = 192$.

Input #2

Output #2

3

Doing zero operations makes the product of the integers on the blackboard $3$.

Input #3

Output #3

824552442

The maximum possible product of the integers on the blackboard after your operations is $5856458868470016$, which should be printed modulo $998244353$.

API Response (JSON)

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  "problem": {
    "name": "Floor, Ceil - Decomposition",
    "description": {
      "content": "There is an integer $X$ written on a blackboard. You can do the operation below any number of times (possibly zero). *   Choose an integer $x$ written on the blackboard. *   Erase one $x$ from the bl",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc135_a"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is an integer $X$ written on a blackboard. You can do the operation below any number of times (possibly zero).\n\n*   Choose an integer $x$ written on the blackboard.\n*   Erase one $x$ from the bl...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments