NOIP2005 循环 题解

算法

(高精度,数学,数论,递推)

#include <cstdio>
#include <cstring>
#include <iostream>
#include <algorithm>

using namespace std;

const int N = 110;

int m;
int nums[N], power[N]; // power存的是n^L_{k-1}

// 高精度乘法
void mul(int c[], int a[], int b[])
{
    static int temp[N];
    memset(temp, 0, sizeof temp);
    for (int i = 0; i < m; i++)
        for (int j = 0; j < m; j++)
            if (i + j < m)
                temp[i + j] += a[i] * b[j];

    for (int i = 0, t = 0; i < m; i++)
    {
        t += temp[i];
        temp[i] = t % 10;
        t /= 10;
    }

    memcpy(c, temp, sizeof temp);
}

// 高精度乘低精度
void mul(int c[], int a[], int b)
{
    for (int i = 0, t = 0; i < m; i++)
    {
        t += a[i] * b;
        c[i] = t % 10;
        t /= 10;
    }
}

int main()
{
    string str;
    cin >> str >> m;
    for (int i = 0, j = str.size() - 1; j >= 0; i++, j--)
        nums[i] = str[j] - '0'; // 把所有的数存入nums里

    memcpy(power, nums, sizeof nums);

    int per[N] = {1}; // per存周期长度
    int pn[N], p1[N];// pn存n*n^iL_{k-1}，p1存1*n^iL_{k-1}
    for (int k = 1; k <= m; k++)
    {
        memcpy(pn, nums, sizeof pn); // 把 n copy到pn里
        memset(p1, 0, sizeof p1); // p1初始化
        p1[0] = 1;

        // 求一下当前的循环节是多少
        int r = 0;
        while (r <= 10)
        {
            mul(pn, pn, power); // pn = pn * power
            mul(p1, p1, power); // p1 = p1 * power
            r++;
            if (pn[k - 1] == nums[k - 1])
                break;
        }

        memcpy(power, p1, sizeof p1);

        if (r > 10)
        {
            memset(per, 0, sizeof per);
            per[0] = -1;
            break;
        }

        mul(per, per, r); // per = per * r
    }

    int k = m;
    while (!per[k])
        k--; //把前面的0去掉

    while (k >= 0)
        cout << per[k--]; // 从高到低把所有位输出

    return 0;
}

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