digitnum

AtCoder

IDabc238_c

Time2000ms

Memory256MB

Difficulty—

Given an integer $N$, solve the following problem. Let $f(x)=$ (The number of positive integers at most $x$ with the same number of digits as $x$). Find $f(1)+f(2)+\dots+f(N)$ modulo $998244353$. ## Constraints * $N$ is an integer. * $1 \le N < 10^{18}$ ## Input Input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

Output #1

73

*   For a positive integer $x$ between $1$ and $9$, the positive integers at most $x$ with the same number of digits as $x$ are $1,2,\dots,x$.
    *   Thus, we have $f(1)=1,f(2)=2,...,f(9)=9$.
*   For a positive integer $x$ between $10$ and $16$, the positive integers at most $x$ with the same number of digits as $x$ are $10,11,\dots,x$.
    *   Thus, we have $f(10)=1,f(11)=2,...,f(16)=7$.

The final answer is $73$.

Input #2

Output #2

Input #3

999999999999999999

Output #3

762062362

Be sure to find the sum modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "digitnum",
    "description": {
      "content": "Given an integer $N$, solve the following problem. Let $f(x)=$ (The number of positive integers at most $x$ with the same number of digits as $x$).   Find $f(1)+f(2)+\\dots+f(N)$ modulo $998244353$.",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc238_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given an integer $N$, solve the following problem.\nLet $f(x)=$ (The number of positive integers at most $x$ with the same number of digits as $x$).  \nFind $f(1)+f(2)+\\dots+f(N)$ modulo $998244353$.\n\n#...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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