Codeforces 933 B. A Determined Cleanup（数学）

Posted 2022-08-13 小坏蛋_千千

Description

In order to put away old things and welcome a fresh new year, a thorough cleaning of the house is a must.

Little Tommy finds an old polynomial and cleaned it up by taking it modulo another. But now he regrets doing this…

Given two integers $p$ and $k$ , find a polynomial $f(x)$ with non-negative integer coefficients strictly less than $k$ , whose remainder is $p$ when divided by $(x + k)$ . That is, $f(x) = q(x)·(x + k) + p$ , where $q(x)$ is a polynomial (not necessarily with integer coefficients).

 

Input

The only line of input contains two space-separated integers $p$ and $k$ $(1 ≤ p ≤ 10^18, 2 ≤ k ≤ 2 000)$ .

 

Output

If the polynomial does not exist, print a single integer $-1$ , or output two lines otherwise.

In the first line print a non-negative integer $d$ — the number of coefficients in the polynomial.

In the second line print $d$ space-separated integers $a_0, a_1, ..., a_d - 1$ , describing a polynomial $f(x)=\\sum_i=0^d-1a_i·x^i$ fulfilling the given requirements. Your output should satisfy $0 ≤ a_i < k$ for all $0 ≤ i ≤ d - 1$ , and $a_d - 1 ≠ 0$ .

If there are many possible solutions, print any of them.

 

Examples input

46 2

 

Examples output

7
0 1 0 0 1 1 1

 

题意

给定正整数 $p,k$ ，我们需要找出一个多项式 $q(x)$ ，使得 $f(x) = q(x)·(x + k) + p$ 的各项系数严格小于 $k$ 且不为负数，输出 $f(x)$ 的各项系数。

 

思路

仔细想想便可以知道，这样的解一定是存在的，所以没有输出 $-1$ 的可能性

我们可以对原式化简一下： $f(x)=x·q(x)+k·q(x)+p$

假设 $q(x)=a_1+a_2·x+a_3·x^2+\\dots+a_n·x^n-1$ ，其中 $a_i$ 为某一整数

现在我们再来讨论 Codeforces 933 C. A Colourful Prospect （平面图，欧拉公式）