Algorithm Gossip: 最大公因数、最小公倍数、因式分解

说明

最大公因数使用辗转相除法来求，最小公倍数则由这个公式来求：
GCD * LCM = 两数乘积

解法

最大公因数可以使用递迴与非递迴求解，因式分解基本上就是使用小于输入数的数值当作除数，去除以输入数值，如果可以整除就视为因数，要比较快的解法就是求出小于该数的所有质数，并试试看是不是可以整除，求质数的问题是另一个课题，请参考 Eratosthenes 筛选求质数。

实作（最大公因数、最小公倍数）

#include <stdio.h> 
#include <stdlib.h> 

int main(void) { 
    int m, n, r; 
    int s;

    printf("输入两数："); 
    scanf("%d %d", &m, &n); 
    s = m * n;

    while(n != 0) { 
        r = m % n; 
        m = n; 
        n = r; 
    } 

    printf("GCD：%d\n", m); 
    printf("LCM：%d\n", s/m); 

    return 0; 
} 

public class GcdLcm {
    public static int gcdOf(int m, int n) {
        int r;
        
        while(n != 0) { 
            r = m % n; 
            m = n; 
            n = r; 
        }
        
        return m;
    }
    
    public static int lcmOf(int m, int n) {
        return m * n / gcdOf(m, n);
    }
    
    public static void main(String[] args) {
        System.out.println("GCD of (10, 4) = " + 
                              GcdLcm.gcdOf(10, 4));
        System.out.println("LCM of (10, 4) = " + 
                              GcdLcm.lcmOf(10, 4));
    }
}

实作（因式分解）

#include <stdio.h> 
#include <stdlib.h> 

int main(void) { 
    int i, n; 

    printf("请输入整数："); 
    scanf("%d", &n); 
    printf("%d = ", n); 

    for(i = 2; i * i <= n;) { 
        if(n % i == 0) { 
            printf("%d * ", i); 
            n /= i; 
        } 
        else 
            i++; 
    } 

    printf("%d\n", n); 

    return 0; 
} 

#include <stdio.h> 
#include <stdlib.h> 

#define N 1000 

int prime(int*);  // 求质数表 
void factor(int*, int);  // 求factor 

int main(void) { 
    int ptable[N+1] = {0}; 
    int count, i, temp; 

    count = prime(ptable); 

    printf("请输入一数："); 
    scanf("%d", &temp); 
    factor(ptable, temp); 

    printf("\n"); 

    return 0; 
 } 

 int prime(int* pNum) { 
    int i, j; 
    int prime[N+1]; 

    for(i = 2; i <= N; i++) 
        prime[i] = 1; 

    for(i = 2; i*i <= N; i++) { 
        if(prime[i] == 1) { 
            for(j = 2*i; j <= N; j++) { 
                if(j % i == 0) 
                    prime[j] = 0; 
            } 
        } 
    } 

    for(i = 2, j = 0; i < N; i++) { 
        if(prime[i] == 1) 
            pNum[j++] = i; 
    } 

    return j; 
} 

void factor(int* table, int num) { 
    int i; 

    for(i = 0; table[i] * table[i] <= num;) { 
        if(num % table[i] == 0) { 
            printf("%d * ", table[i]); 
            num /= table[i]; 
        } 
        else 
            i++; 
    } 

    printf("%d\n", num); 
}
 

import java.util.ArrayList;

public class Factor {
    public static int[] factor(int num) {
        int[] pNum = Prime.findPrimes(num);
        
        ArrayList list = new ArrayList();
        
        for(int i = 0; pNum[i] * pNum[i] <= num;) { 
            if(num % pNum[i] == 0) { 
                list.add(new Integer(pNum[i]));
                num /= pNum[i]; 
            } 
            else 
                i++; 
        } 

        list.add(new Integer(num));
        
        int[] f = new int[list.size()];
        Object[] objs = list.toArray();
        for(int i = 0; i < f.length; i++) {
            f[i] = ((Integer) objs[i]).intValue();
        }
        
        return f;
    }
    
    public static void main(String[] args) {
        int[] f = Factor.factor(100);
        for(int i = 0; i < f.length; i++) {
            System.out.print(f[i] + " ");
        }
        System.out.println();
    }
}