算法比赛中常用的数学知识

一、求某个整数的正约数个数与正约数之和

1.1求某个正整数N的正约数个数 

public class Main {
    public static void main(String[] args) {
        System.out.println(count(360));//结果为24
    }
    public static long count(long number){
        long count=1;
        for(long i=2;i<=Math.sqrt(number);i++){
            long n=0;
            while (number%i==0){
                number=number/i;
                n++;
            }
            if(n>0){
                count=count*(n+1);
            }
        }
        if(number>1){
            count=count*2; //处理的是1次幂的情况，因为（1+1）=2；
        }
        return count;
    }
}

 1.2求某个正整数N的正约数之和 

public class Main {
    public static void main(String[] args) {
        System.out.println(count(360));
    }

    public static long count(long number) {
        long sum = 1;
        for (long i = 2; i <= Math.sqrt(number); i++) {
            long exponent = 0;
            while (number % i == 0) {
                number /= i;
                exponent++;
            }
            if (exponent > 0) {
                // 正确使用等比数列求和公式计算某个质因数对应的约数和
                long termSum = (long) ((Math.pow(i, exponent + 1) - 1) / (i - 1));
                // 累乘不同质因数对应的约数和
                sum *= termSum;
            }
        }
        //处理指数为1的情况，因为number的1次方是number,0次方是1
        if (number > 1) {
            sum *= (number + 1);
        }
        return sum;
    }
}

二、求最大公约数

//求最大公约数
    public static long gcd(long a,long b) {
    	if(b==0) {
    		return a;
    	}
    	return gcd(b,a%b);
    	
    	
    }

三、分解质因数

 public static void ps(int n) {
    	List<Integer> list=new ArrayList<>();
    	for(int i=2;i<=Math.sqrt(n);i++) {
    		while(n%i==0) {
    			list.add(i);  //找到一个质因数
    			n=n/i;  //更新待分解的数
    		}
    		
    	}
    	if(n>1) {
			list.add(n); //如果最后余数大于1，则说明它本身也是质数
		}
}

四、快速幂

import java.util.Scanner;

public class a {
    public static void main(String[] args) {
        Scanner scan=new Scanner(System.in);
        long b=scan.nextInt();
        long p=scan.nextInt();
        long k=scan.nextInt();
        System.out.println(quickmi(b,p,k));

    }
    public static long quickmi(long b,long p,long k){
        long res=1;
        while(p>0){
            if((p&1)==1){
                res=res*b%k;
            }
            b=b*b%k;
            p=p>>1;

        }
        return res;
    }

}

 五、费马小定理

乘法逆元的定义

 

import java.util.Scanner;

public class a {
    public static void main(String[] args) {
        Scanner scan = new Scanner(System.in);
        int n = scan.nextInt();
        while (n-- > 0) {
            long b=scan.nextLong();
            long p=1000000007-2;
            long k=1000000007;
            System.out.println(quickmi(b,p,k));
        }
    }
    public static long quickmi(long b,long p,long k){
        long res=1;
        while(p>0){
            if((p&1)==1){
                res=res*b%k;
            }
            b=b*b%k;
            p=p>>1;

        }
        return res;
    }

}

五，求组合数问题(注意看数据范围，不同范围不同求法）

 

import java.util.Scanner;

public class Main {

    static long[] arr=new long[16];
    public static void main(String[] args) {
        Scanner scan=new Scanner(System.in);
        int q=scan.nextInt();
        arr[0]=1;
        arr[1]=1;
        for(int i=2;i<16;i++) {
            arr[i]=i*arr[i-1];
        }
        while(q-->0) {
            int n=scan.nextInt();
            int m=scan.nextInt();
            System.out.println(jiec(n,m));
        }
    }
    public static long jiec(int n,int m) {
        long ans=arr[n]/(arr[m]*arr[n-m]);
        return ans;
    }

}    

用快速幂加费马小定理

import java.util.Scanner;

public class Main {
    static long N=1000000007;
    static long[] arr;
    public static void main(String[] args) {
        Scanner scan=new Scanner(System.in);
        int a=scan.nextInt();
        int b=scan.nextInt();
        arr=new long[3001];
        arr[0]=1;
        for(int i=1;i<3001;i++){
            arr[i]=arr[i-1]*i%N;
        }
        System.out.println(jisuan(a,b));
    }
    //快速幂,a的b次方
    public static long quickmi(long a,long b){
        long ans=1;
        while(b>0){
            if((b&1)==1){
                ans=ans*a%N;
            }
            a=a*a%N;
            b=b>>1;
        }
        return ans;
    }
    //计算组合数
   public static long jisuan(int a,int b){
        long fenmu=arr[b]*arr[a-b]%N;
        //费马小定理
        long sum=quickmi(fenmu,N-2);
        long ans=arr[a]*sum%N;
        return ans;
   }
}

六、素数筛

import java.util.ArrayList;
import java.util.List;
import java.util.Scanner;

public class a {
    public static void main(String[] args) {
        Scanner scan = new Scanner(System.in);
        int n = scan.nextInt();
        boolean[] prime = new boolean[n + 1]; //标记是否为素数
        for (int i = 2; i <= n; i++) {
            prime[i] = true;
        }
        List<Integer> list = new ArrayList<>();  //存素数
        for (int i = 2; i <= n; i++) {
            if (prime[i]) {
                list.add(i);
            }
            for (int j = 0; j < list.size() && i * list.get(j) <= n; j++) {
                //i * list.get(j) <= n这个条件是为了防止数组越界
                prime[i * list.get(j)] = false;
                if (i % list.get(j) == 0) {
                    break;
                }
            }
        }
        System.out.println(list.size());
    }

}

七、求三角形面积