快速幂

快速幂算法例子:​​

$$5^{16}=25^8=625^4=390625^2=152 587 890 625​$$
$$3^{20}=9^{10}=81^5=81*81^4=81*6561^2=81*43046721=3486784401$$

利用二分法能大大加快了求解速度

代码:

int Power(int base,int index){
    int result=1;
    while(index!=0){
        if(index%2 == 1) result =result*base;
        base = base* base;
        index/=2;
    } 
    return result;
}

快速幂求余:

​$$3^{20}\%4=(9\%4)^{10}\%4=5^{10}\%4=(25\%4)^5\%4=1^5\%4 \Rightarrow result=(1*1)\%4=1$$
$$6^{13}\%7 \Rightarrow result =(1*6)\%7=6,(36\%7)^6\%7=(1\%7)^3 \Rightarrow result=(6*1)\%7=6,1\%7....result=6 $$

int PowerMod(int base, int index, int mod){  
    int result = 1;  
    base = base % mod;  
    while(index!=0){    
        if(index%2 == 1)  result =(result*base)%mod;  
        base = (base * base) % mod;  
        index /= 2;
    }  
    return result;  
}