数论基础 (gcd+lcm+分解质因数+欧拉函数+欧拉定理+费马小定理+扩展欧几里得)模板

gcd+lcm

#include<bits/stdc++.h>
using namespace std;
typedef long long LL;

LL gcd(LL a,LL b){
  return b==0?a:gcd(b,a%b);
}

LL lcm(LL a,LL b){
  return a*b/gcd(a,b);
}
int main(){
  int n;
  LL a,b;
  scanf("%d",&n);
  while(n--){
    scanf("%lld %lld",&a,&b);
    printf("%lld %lld\n",gcd(a,b),lcm(a,b));
  }
  return 0;
}

分解质因数

#include<bits/stdc++.h>
using namespace std;
typedef long long LL;
int main(){
  LL x;
  int n;
  scanf("%d",&n);
  while(n--){
    scanf("%lld",&x);
    printf("%lld = ",x);
    for(int i=2;i*i<=x;i++){
      if(x%i==0){
        int cnt = 0;
        while(x%i==0){
          x=x/i;
          cnt++;
        }
        printf("%d",i);
        if(cnt>1) printf("^%d",cnt);
        if(x!=1) printf(" * ");
      }
    }
    if(x!=1) {
      printf("%lld\n",x);
    }else if(x==1){
      printf("\n");
    }
  }
  return 0;
}

费马小定理+快速幂

#include<iostream>
#include<cstdio>
using namespace std;
typedef long long LL;
const int p=19260817;

LL poww(LL a,LL n,LL p){
  LL res = 1;
  while(n){
    if(n & 1) res = (res * a) % p;
    a = (a * a) % p;
    n >>= 1;
  }
  return res;
}

int main(){
    int T;
    cin>>T;
    while(T--){
        LL n;
        cin>>n;

        cout<<poww(n,p-2,p)<<endl;
    }
    return 0;
}

扩展欧几里得

#include<bits/stdc++.h>
using namespace std;
typedef long long LL;
const LL p=19260817;
void exgcd(LL a,LL b,LL &x,LL &y){
  if(b == 0){
    x = 1;y = 0;
  }else{
    exgcd(b,a%b,y,x);
    y -= (a/b) * x;
  }
}

int main(){
  int t;
  LL x,y;
  cin>>t;
  while(t--){
    LL n;
    cin>>n;
    exgcd(n,p,x,y);
    cout<<(x%p+p)%p<<endl;
    //cout<<(x+p)%p<<endl;//或者
    //由于扩欧的结果只满足|x|+|y|最小，由题目的要求x为正整数应该输出(x%b+b)%b（由于x+b可能还是小于0）
  }
  return 0;
}

欧拉函数

#include<bits/stdc++.h>
using namespace std;
typedef long long LL;

LL phi(LL x){
  LL ans =x;
  for(LL pi=2;pi*pi<=x;pi++){
    if(x%pi==0){
      ans=(ans/pi)*(pi-1);
      while(x%pi==0) x/=pi;
    }
  }
  if(x>1) ans=(ans/x)*(x-1);
  return ans;
}

int main(){
  int t;
  cin>>t;
  LL a;
  while(t--){
    scanf("%lld",&a);
    printf("%lld\n",phi(a));
  }
  return 0;
}

欧拉定理

#include<bits/stdc++.h>
using namespace std;
typedef long long LL;

LL Eular(LL x){
  LL phi =x;
  for(LL pi=2;pi*pi<=x;pi++){
    if(x%pi==0){
      phi=(phi/pi)*(pi-1);
      while(x%pi==0) x/=pi;
    }
  }
  if(x>1) phi=(phi/x)*(x-1);
  return phi;
}

LL gcd(LL a,LL b){
  return b==0?a:gcd(b,a%b);
}

LL pow(int a,int x,int mod){
    LL ans=1;
    for(int i=0;i<x;i++){
        ans=ans*a%mod;
    }
    return ans;
}

int main(){
  LL t,a,c,n;
  cin>>t;
  while(t--){
    scanf("%lld%lld%lld",&a,&c,&n);
    LL p = Eular(n);
    if(gcd(a,n)==1){
      printf("%lld\n",pow(a,c%p,n));
    }else{
      if(c<p){
        printf("%lld\n",pow(a,c,n));
      }else if(c>=p){
        printf("%lld\n",pow(a,c%p+p,n));
      }
    }
  }
  return 0;
}