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Minimum value exceeding X whose count of divisors has different parity with count of divisors of X
  • Difficulty Level : Expert
  • Last Updated : 07 Dec, 2020

Given an integer X, the task is to determine the minimum value of Y greater than X, such that count of divisors of X and Y have different parities.

Examples:

Input: X = 5
Output: 9
Explanation: The count of divisors of 5 and 9 are 2 and 3 respectively, which are of different parities.

Input: X = 9
Output: 10
Explanation: The counts of divisors of 9 and 10 are 3 and 4, which are of different parities.

 

Naive Approach: The simplest approach to solve the problem is to iterate each number starting from X + 1 until an element with count of the divisors with parity opposite to that of X is obtained.



Time Complexity: O((1+√X)2)
Auxiliary Space: O(1)

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to count divisors of n
int divisorCount(int n)
{
    int x = 0;
    for (int i = 1; i <= sqrt(n); i++) {
        if (n % i == 0) {
            if (i == n / i)
                x++;
            else
                x += 2;
        }
    }
    return x;
}
 
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
int minvalue_y(int x)
{
    // Divisor count of x
    int a = divisorCount(x);
    int y = x + 1;
 
    // Iterate from x + 1 and
    // check for each element
    while ((a & 1)
           == (divisorCount(y) & 1))
        y++;
    return y;
}
 
// Driver Code
int main()
{
    // Given X
    int x = 5;
 
    // Function call
    cout << minvalue_y(x) << endl;
    return 0;
}

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Java

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// Java program for the above approach
import java.util.*;
 
class GFG{
 
// Function to count divisors of n
static int divisorCount(int n)
{
    int x = 0;
    for(int i = 1; i <= Math.sqrt(n); i++)
    {
        if (n % i == 0)
        {
            if (i == n / i)
                x++;
            else
                x += 2;
        }
    }
    return x;
}
 
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
static int minvalue_y(int x)
{
     
    // Divisor count of x
    int a = divisorCount(x);
    int y = x + 1;
 
    // Iterate from x + 1 and
    // check for each element
    while ((a & 1) == (divisorCount(y) & 1))
        y++;
         
    return y;
}
 
// Driver Code
public static void main(String[] args)
{
     
    // Given X
    int x = 5;
 
    // Function call
    System.out.println(minvalue_y(x));
}
}
 
// This code is contributed by chitranayal

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C#

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// C# program for the above approach
using System;
  
class GFG{
  
// Function to count divisors of n
static int divisorCount(int n)
{
    int x = 0;
    for(int i = 1; i <= Math.Sqrt(n); i++)
    {
        if (n % i == 0)
        {
            if (i == n / i)
                x++;
            else
                x += 2;
        }
    }
    return x;
}
  
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
static int minvalue_y(int x)
{
      
    // Divisor count of x
    int a = divisorCount(x);
    int y = x + 1;
  
    // Iterate from x + 1 and
    // check for each element
    while ((a & 1) == (divisorCount(y) & 1))
        y++;
          
    return y;
}
  
// Driver Code
public static void Main()
{
      
    // Given X
    int x = 5;
  
    // Function call
    Console.WriteLine(minvalue_y(x));
}
}
 
// This code is contributed by susmitakundugoaldanga

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Python3

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# Python program for the above approach
 
# Function to count divisors of n
def divisorCount(n):
    x = 0;
    for i in range(1, n):
        if (n % i == 0):
            if (i == n // i):
                x += 1;
            else:
                x += 2;
        if(i * i > n):
            break;
 
    return x;
 
# Function to find the minimum
# value exceeding x whose count
# of divisors has different parity
# with count of divisors of X
def minvalue_y(x):
   
    # Divisor count of x
    a = divisorCount(x);
    y = x + 1;
 
    # Iterate from x + 1 and
    # check for each element
    while ((a & 1) == (divisorCount(y) & 1)):
        y += 1;
 
    return y;
 
# Driver Code
if __name__ == '__main__':
   
    # Given X
    x = 5;
 
    # Function call
    print(minvalue_y(x));
 
# This code is contributed by 29AjayKumar

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Output: 

9

 

Efficient Approach: The problem can be solved based on the following observations:

Follow the steps below to solve the problem:

  1. Check if X is a perfect square. If found to be true, print X + 1.
  2. Otherwise, print (1 + floor(√X))2).

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
int minvalue_y(int x)
{
    // Check if x is
    // perfect square
    int n = sqrt(x);
    if (n * n == x)
        return x + 1;
 
    return pow(n + 1, 2);
}
 
// Driver Code
int main()
{
    int x = 5;
    cout << minvalue_y(x) << endl;
    return 0;
}

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Java

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// Java program for the above approach
import java.util.*;
    
class GFG{
    
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
static int minvalue_y(int x)
{
     
    // Check if x is
    // perfect square
    int n = (int)Math.sqrt(x);
    if (n * n == x)
        return x + 1;
  
    return (int)Math.pow(n + 1, 2);
}
   
// Driver Code
public static void main(String[] args)
{
    int x = 5;
      
    System.out.print(minvalue_y(x));
}
}
 
// This code is contributed by sanjoy_62

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Python3

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# Python3 program for the above approach
 
# Function to find the minimum
# value exceeding x whose count
# of divisors has different parity
# with count of divisors of X
def minvalue_y(x):
     
    # Check if x is
    # perfect square
    n = int(pow(x, 1 / 2))
     
    if (n * n == x):
        return x + 1
         
    return(pow(n + 1, 2))
 
# Driver Code
if __name__ == '__main__':
     
    x = 5
 
    print(minvalue_y(x))
 
# This code is contributed by Rajput-Ji

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C#

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// C# program for the above approach
using System;
     
class GFG{
     
// Function to find the minimum
// value exceeding x whose count
// of divisors has different parity
// with count of divisors of X
static int minvalue_y(int x)
{
     
    // Check if x is
    // perfect square
    int n = (int)Math.Sqrt(x);
    if (n * n == x)
        return x + 1;
         
    return (int)Math.Pow(n + 1, 2);
}
    
// Driver Code
public static void Main()
{
    int x = 5;
     
    Console.WriteLine(minvalue_y(x));
}
}
 
// This code is contributed by code_hunt

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Output: 

9

 

Time Complexity: O(1)
Auxiliary Space: O(1)

 

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