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C - digitnum

Score : $300$ points

Problem Statement

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:

$N$

Output

Print the answer as an integer.

16

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 $73$.

238

13870

999999999999999999

762062362

Be sure to find the sum modulo $998244353$.