检查数的奇数和偶数的计数是否相等

给定数字N ，任务是查找N是否具有相等数量的奇数和偶数因子。
例子：

Input: N = 10 
Output: YES 
Explanation: 10 has two odd factors (1 and 5) and two even factors (2 and 10)
Input: N = 24 
Output: NO 
Explanation: 24 has two odd factors (1 and 3) and six even factors (2, 4, 6, 8 12 and 24)
Input: N = 125 
Output: NO 
 

天真的方法：查找所有除数，并检查奇数除数的数量是否与偶数除数的数量相同。
下面是上述方法的实现

C++

// C++ solution for the above approach
#include 
using namespace std;
#define lli long long int
 
// Function to check condition
void isEqualFactors(lli N)
{
    // Initialize even_count
    // and od_count
    lli ev_count = 0, od_count = 0;
 
    // loop runs till square root
    for (lli i = 1;
         i <= sqrt(N) + 1; i++) {
 
        if (N % i == 0) {
            if (i == N / i) {
                if (i % 2 == 0)
                    ev_count += 1;
                else
                    od_count += 1;
            }
 
            else {
                if (i % 2 == 0)
                    ev_count += 1;
                else
                    od_count += 1;
 
                if ((N / i) % 2 == 0)
                    ev_count += 1;
 
                else
                    od_count += 1;
            }
        }
    }
 
    if (ev_count == od_count)
        cout << "YES" << endl;
    else
        cout << "NO" << endl;
}
 
// Driver Code
int main()
{
    lli N = 10;
    isEqualFactors(N);
 
    return 0;
}

Java

// Java solution for the above approach
class GFG{
 
// Function to check condition
static void isEqualFactors(int N)
{
     
    // Initialize even_count
    // and od_count
    int ev_count = 0, od_count = 0;
 
    // Loop runs till square root
    for(int i = 1;
            i <= Math.sqrt(N) + 1; i++)
    {
       if (N % i == 0)
       {
           if (i == N / i)
           {
               if (i % 2 == 0)
                   ev_count += 1;
               else
                   od_count += 1;
           }
           else
           {
               if (i % 2 == 0)
                   ev_count += 1;
               else
                   od_count += 1;
                    
               if ((N / i) % 2 == 0)
                   ev_count += 1;
               else
                   od_count += 1;
           }
       }
    }
     
    if (ev_count == od_count)
        System.out.print("YES" + "\n");
    else
        System.out.print("NO" + "\n");
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 10;
     
    isEqualFactors(N);
}
}
 
// This code is contributed by amal kumar choubey

Python3

# Python3 code for the naive approach
import math
 
# Function to check condition
def isEqualFactors(N):
 
# Initialize even_count
# and od_count
    ev_count = 0
    od_count = 0
 
# loop runs till square root
    for i in range(1, (int)(math.sqrt(N)) + 2) :
        if (N % i == 0):
            if(i == N // i):
                if(i % 2 == 0):
                    ev_count += 1
                else:
                    od_count += 1
             
            else:
                if(i % 2 == 0):
                    ev_count += 1
                else:
                    od_count += 1
                     
                if((N // i) % 2 == 0):
                    ev_count += 1;
                else:
                    od_count += 1;
     
    if (ev_count == od_count):
        print("YES")
    else:
        print("NO")
 
# Driver Code
N = 10
isEqualFactors(N)

C#

// C# solution for the above approach
using System;
 
class GFG{
 
// Function to check condition
static void isEqualFactors(int N)
{
     
    // Initialize even_count
    // and od_count
    int ev_count = 0, od_count = 0;
 
    // Loop runs till square root
    for(int i = 1;
            i <= Math.Sqrt(N) + 1; i++)
    {
        if (N % i == 0)
        {
            if (i == N / i)
            {
                if (i % 2 == 0)
                    ev_count += 1;
                else
                    od_count += 1;
            }
            else
            {
                if (i % 2 == 0)
                    ev_count += 1;
                else
                    od_count += 1;
                         
                if ((N / i) % 2 == 0)
                    ev_count += 1;
                else
                    od_count += 1;
            }
        }
    }
     
    if (ev_count == od_count)
        Console.Write("YES" + "\n");
    else
        Console.Write("NO" + "\n");
}
 
// Driver Code
public static void Main(String[] args)
{
    int N = 10;
     
    isEqualFactors(N);
}
}
 
// This code is contributed by gauravrajput1

输出：

YES

时间复杂度： O(sqrt(N))
辅助空间： O(1)
高效的解决方案：
必须进行以下观察以优化上述方法：

• 根据唯一因式分解定理， 质数的乘积可以表示任何数字。因此，N可以表示为：

 
 

N = P₁^A₁ * P₂^A₂ * P₃^A₃ * …….. * P_k^A_K where, each P_i is a prime and each A_i is a positive integer.(1 <= i <= K) 
 

 

• 根据组合定律，N的任何除数都可以采用以下形式：

N = P₁^B₁ * P₂^B₂ * P₃^B₃ * …….. * P_K^B_K where B_i is an integer and 0 <= B_i <= A_i for 1 <= i <= K. 
 

• 如果除数在素数分解中不包含2，则将是奇数。因此，如果P ₁ = 2，则B ₁必须为0 。它只能以一种方式完成。
• 对于偶数除数，可以将B ₁替换为1(或)2(或).. A ₁以得到除数。可以用B ₁种方式完成。
• 现在，对于其他人，每个B _i都可以用0(或)1(或)2…..(或)A _i替换为1 <= i <=K。这可以用(A _i +1)方式完成。
• 根据基本原则：
  • 奇数除数的数量为： X = 1 *(A ₂ +1)*(A ₃ +1)*….. *(A _K +1)。
  • 同样，偶数除数为： Y = A ₁ *(A ₂ +1)*(A ₃ +1)*…。 *(A _K +1)。
• 不可以除数的等于等于no。 X，Y的奇数除数应该相等。仅当A ₁ = 1时才有可能。

因此，可以得出结论，如果一个数的偶数和奇数除数在素因数分解中具有1(只有1)的2的幂，则该数相等。
请按照以下步骤解决问题：

1. 对于给定的数字N，检查它是否可以被2整除。
2. 如果数字可被2整除，则检查数字是否可被2 ²整除。如果是，则该数字将不具有相等数量的奇偶数。如果不是，则该数字将具有相等数量的奇数和偶数因子。
3. 如果该数字不能被2整除，则该数字将永远不会有偶数因数，因此它的奇数和偶数因数也将不相等。

下面是上述方法的实现。

C

// C code for the above approach
#include 
#define lli long long int
 
// Function to check condition
void isEqualFactors(lli N)
{
    if ((N % 2 == 0)
        && (N % 4 != 0))
        printf("YES\n");
    else
        printf("NO\n");
}
 
// Driver Code
int main()
{
    lli N = 10;
    isEqualFactors(N);
 
    N = 125;
    isEqualFactors(N);
 
    return 0;
}

C++

// C++ code for the above approach
#include 
using namespace std;
#define lli long long int
 
// Function to check condition
void isEqualFactors(lli N)
{
    if ((N % 2 == 0)
        and (N % 4 != 0))
        cout << "YES" << endl;
    else
        cout << "NO" << endl;
}
 
// Driver Code
int main()
{
    lli N = 10;
    isEqualFactors(N);
 
    N = 125;
    isEqualFactors(N);
 
    return 0;
}

Java

// JAVA code for the above approach
public class GFG {
 
    // Function to check condition
    static void isEqualFactors(int N)
    {
        if ((N % 2 == 0)
            && (N % 4 != 0))
            System.out.println("YES");
 
        else
            System.out.println("NO");
    }
 
    // Driver code
    public static void main(String args[])
    {
        int N = 10;
        isEqualFactors(N);
 
        N = 125;
        isEqualFactors(N);
    }
}

Python

# Python code for the above approach
  
# Function to check condition
def isEqualFactors(N):
    if ( ( N % 2 == 0 ) and ( N % 4 != 0) ):
        print("YES")
    else:
        print("NO")
 
# Driver Code
N = 10
isEqualFactors(N)
 
N = 125;
isEqualFactors(N)

C＃

// C# code for the above approach
using System;
 
class GFG {
 
    // Function to check condition
    static void isEqualFactors(int N)
    {
        if ((N % 2 == 0)
            && (N % 4 != 0))
            Console.WriteLine("YES");
 
        else
            Console.WriteLine("NO");
    }
 
    // Driver code
    public static void Main()
    {
        int N = 10;
        isEqualFactors(N);
 
        N = 125;
        isEqualFactors(N);
    }
}

Java脚本

 
// Javascript code for the above approach
 
// Function to check condition
function isEqualFactors(N)
{
    if ((N % 2 == 0)
        && (N % 4 != 0))
        document.write("YES<br>");
    else
        document.write("NO<br>");
}
 
// Driver Code
var N = 10;
isEqualFactors(N);
N = 125;
isEqualFactors(N);
 

输出：

YES
NO

时间复杂度： O(1)
辅助空间： O(1)