P12397 「FAOI-R9」Function Master

As a computer technology master Mingyue likes using Geometer's Sketchpad to draw the graphs of various bizarre functions, like $ y=\frac{x^x}{\sin x} $, $ y=\lfloor x^{\tan x} \rfloor $, and $ y=\frac{x+x^3+x^5+x^7}{1+x^2+x^4+x^6}$. Some of them are continuous, some are discrete, and some have very strange shapes. However, as a math master who scored 99/100 in the high school entrance examination in math, he is confident that he can grasp the laws of many functions. So, Qingfeng give him such a function.

Qingfeng denotes a function $s(x)\;(x \in \mathbb N^*)$ as the sum of each digit of $x$ in decimal. Formally, $s(x)$ is defined as: $$ s(x) = \sum_{i=0} \left( \lfloor \frac{x}{10^i} \rfloor \bmod 10 \right). $$ Consequently, he defines $S_k(x)\;(x \in \mathbb N^*, k \in \mathbb N)$ as follows: $$ S_k(x) = \begin{cases} x, & k = 0, \\ s(S_{k-1}(x)), & k > 0. \end{cases} $$ Moreover, Qingfeng defines $f_k(x)\;(x \in \mathbb N^*, k \in \mathbb N)$ to be $\sum_{i=0}^k S_i(x)$. After giving such a function to Mingyue, who confidently inputs the formula into the Geometer's Sketchpad and is dazzled by the graphics, Mingyue decides to ask you to find out the properties of the function. To be specific: you are given a constant integer $k$. There are multiple queries, in each of them you are given another integer $m$, and you have to find out the number of intersection points of the graphics of the functions $y = f_k(x)$ and $y = m$. It can be proven that the result is always finite under the given constraints.

The first line contains two integers $T$ and $k$, denoting the number of queries, and the given constant, respectively. Then $T$ lines follow. Each line contains a single integer $m$.

For each query, output a single line containing an integer: the answer to the corresponding query.

#### Sample Description For the first test case: the sets of the x - coordinates of all the intersection points corresponding to each group of data are $\{12\}$, $\{5\}$, $\varnothing$, and $\{26\}$, respectively. #### Constraints **Subtasks are used in this problem.** - Subtask 1（5 pts）: $ k=0 $. - Subtask 2（20 pts）: $ T \le 10 $, $ m \le 10^5 $, $ k \le 10 $. - Subtask 3（25 pts）: $ T \le 10 $, $ m \le 10^6 $, $ k \le 10^4 $. - Subtask 4（25 pts）: $ k \le 1 $. - Subtask 5（25 pts）: No additional constraints. For all test cases, it is guaranteed that $1\le T\le 10^5$, $0\le k\le 10^9$, $0\le m\le 10^{18}$.