Codeforces 934.D A Determined Cleanup
1 second
256 megabytes
standard input
standard output
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).
The only line of input contains two space-separated integers p and k (1 ≤ p ≤ 1018, 2 ≤ k ≤ 2 000).
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 a0, a1, ..., ad - 1, describing a polynomial fulfilling the given requirements. Your output should satisfy 0 ≤ ai < k for all 0 ≤ i ≤ d - 1, and ad - 1 ≠ 0.
If there are many possible solutions, print any of them.
46 2
7
0 1 0 0 1 1 1
2018 214
3
92 205 1
In the first example, f(x) = x6 + x5 + x4 + x = (x5 - x4 + 3x3 - 6x2 + 12x - 23)·(x + 2) + 46.
In the second example, f(x) = x2 + 205x + 92 = (x - 9)·(x + 214) + 2018.
题目大意:给定k和p,要求一个多项式f(x) = q(x)(x+k) + p,其中f(x)的每个系数都是非负的,并且小于k.
分析:这道题看着有点难,其实分析出规律来以后挺简单的.
考虑构造q(x).先让常数项小于k,也就是让相乘后的常数项+p小于k.q(x)的常数项就是p / (-k).这样会产生一次项,那么继续消一次项.直到最后的p = 0.
原理其实就是q(x)的第i次项与k相乘,使得第i-1次项与k项乘的结果加上它以后小于k.
#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm> using namespace std; typedef long long ll; ll p,k,ans[],cnt; ll get()
{
ll x = p % k;
if (x < )
x += k;
return x % k;
} int main()
{
cin >> p >> k;
while (p)
{
ans[++cnt] = get();
p -= get();
p /= (-k);
}
cout << cnt << endl;
for (int i = ; i <= cnt; i++)
cout << ans[i] << " "; return ;
}
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