正题

题目链接:https://www.luogu.org/problem/P5091

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题目大意

求$a^b\mathrm{m}\mathrm{o}\mathrm{d}m$

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欧拉定理及扩展

欧拉定理:

对于$(a,m)=1$那么有$$a^{\phi (m)}\equiv 1(modm)$$

扩展欧拉定理

$$a^b\equiv \{\begin{array}{c}{a}^{b\%\phi (p)+\phi (p)}(b>\phi (p))\\ a^{b\%\phi (p)}(b\le \phi (p))\end{array}(modp)$$

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解题思路

然而本题是扩展欧拉定理，直接计算即可。

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$code$

#include<cstdio>
#include<cstring>
#include<cctype>
#include<algorithm>
#define ll long long
using namespace std;
ll a,p,phi,b;
char c;bool flag;
ll power(ll x,ll b,ll p)
{ll ans=1;while(b){if(b&1) ans=ans*x%p;x=x*x%p;b>>=1;}return ans;
}
int main()
{scanf("%lld%lld",&a,&p);phi=p;ll P=p;for(ll i=2;i*i<=p;i++)if(p%i==0){phi=phi*(i-1)/i;while(!(p%i)) p/=i;}if(p>1) phi=phi*(p-1)/p;while(!isdigit(c=getchar()));for(;isdigit(c);c=getchar()){b=b*10+c-'0';if(b>=phi)b%=phi,flag=1;}b+=phi*flag;printf("%lld",power(a,b,P));
}