题解P4884 多少个1
一道比较裸的BSGS题
根据题意得出 \(10^n+10^{n-1}+...+1\equiv k(mod~m)\Rightarrow \frac{10^n-1}{9}\equiv k(mod~m)\Rightarrow 10^n\equiv 9k+1(mod~m)\)
直接套BSGS板子就行了,注意在乘的过程中可能爆long long,加个快速乘,时间复杂度 \(O(\sqrt{n}\log{n})\)
代码
#include<iostream>
#include<cstdio>
#include<unordered_map>
#include<cmath>
using namespace std;
typedef long long ll;
ll ksc(ll a, ll b, ll p)
{
ll res = 0;
while (b)
{
if (b & 1)
res = (res + a) % p;
a = (a + a) % p;
b >>= 1;
}
return res;
}
ll ksm(ll a, ll b, ll p)
{
ll res = 1;
while (b)
{
if (b & 1)
res = ksc(res, a, p);
a = ksc(a, a, p);
b >>= 1;
}
return res;
}
ll BSGS(ll a, ll b, ll p)
{
if (1 % p == b % p)
return 0;
unordered_map<ll, ll> hash;
ll k = sqrt(p) + 1;
for (ll i = 0, j = b % p; i < k; i++, j = ksc(j, a, p))
hash[j] = i;
ll ak = ksm(a, k, p);
for (ll i = 1, j = ak; i <= k; i++, j = ksc(j, ak, p))
if (hash.count(j))
return i * k - hash[j];
return -1;
}
int main()
{
ll k, m;
scanf("%lld%lld", &k, &m);
printf("%lld", BSGS(10, (9 * k + 1) % m, m));
}