D Numbers

D Number is a number N > 3 such that N divides k^(n-2)-k for all k with gcd(k, n) = 1, 1<k<n.

9, 15, 21, 33, 39, 51, 57, 63, 69, 87, 93….

Check if a number is a D-Number

Given a number N, the task is to check if N is an D Number or not. If N is an D Number then print “Yes” else print “No”.

Examples:

Input: N = 9
Output: Yes
Explanation:
9 is a D-number since it divides all the numbers
2^7-2, 4^7-4, 5^7-5, 7^7-7 and 8^7-8, and
2, 4, 5, 7, 8 are relatively prime to n.



Input: N = 16
Output: No

Approach: Since D Number is a number N > 3 such that N divides k^(n-2)-k for all k with gcd(k, n) = 1, 1<k<n. So in a loop of k from 2 to n-1, we will check if gcd of N and k is 1 or not.If it's 1 then we will check if k^(n-2)-k is divisible by N or not.If not divisible we will return false.At last we will return true.

Below is the implementation of the above approach:

C++

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// C++ implementation 
// for the above approach
#include<bits/stdc++.h>
using namespace std;
  
// Function to find the N-th 
// icosikaipentagon number 
int isDNum(int n)
{
    // number should be 
    // greater than 3
    if (n < 4)
        return false;
      
    int numerator, hcf;
      
    // Check every k in range 2 to n-1
    for (int k = 2; k <= n; k++) 
    {
        numerator = pow(k, n - 2) - k;
        hcf = __gcd(n, k);
    }
      
    // condition for D-Number
    if (hcf == 1 && (numerator % n) != 0)
        return false;
      
    return true;
}
  
// Driver Code
int main() 
{
    int n = 15;
    int a = isDNum(n);
    if (a)
        cout << "Yes";
    else
        cout << "No";
}
  
// This code is contributed by Ritik Bansal

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Java

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// Java implementation for the 
// above approach
import java.util.*;
  
class GFG{
  
// Function to find the N-th 
// icosikaipentagon number 
static boolean isDNum(int n)
{
      
    // Number should be 
    // greater than 3
    if (n < 4)
        return false;
      
    int numerator = 0, hcf = 0;
      
    // Check every k in range 2 to n-1
    for(int k = 2; k <= n; k++) 
    {
       numerator = (int)(Math.pow(k, n - 2) - k);
       hcf = __gcd(n, k);
    }
      
    // Condition for D-Number
    if (hcf == 1 && (numerator % n) != 0)
        return false;
      
    return true;
}
  
static int __gcd(int a, int b) 
    return b == 0 ? a : __gcd(b, a % b);     
  
// Driver Code
public static void main(String[] args) 
{
    int n = 15;
    boolean a = isDNum(n);
      
    if (a)
        System.out.print("Yes");
    else
        System.out.print("No");
}
}
  
// This code is contributed by Amit Katiyar

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Python3

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# Python3 implementation 
# for the above approach
  
import math 
    
# Function to find the N-th 
# icosikaipentagon number 
def isDNum(n): 
    # number should be 
        # greater than 3
    if n < 4:
        return False
  
    # Check every k in range 2 to n-1
    for k in range(2, n):
        numerator = pow(k, n - 2) - k
        hcf = math.gcd(n, k)
  
        # condition for D-Number
        if(hcf ==1 and (numerator % n) != 0):
            return False
    return True
  
# Driver code 
n = 15
if isDNum(n):
    print("Yes")
else:
    print("No")

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C#

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// C# implementation for the 
// above approach
using System;
class GFG{
  
// Function to find the N-th 
// icosikaipentagon number 
static bool isDNum(int n)
{
      
    // Number should be 
    // greater than 3
    if (n < 4)
        return false;
      
    int numerator = 0, hcf = 0;
      
    // Check every k in range 2 to n-1
    for(int k = 2; k <= n; k++) 
    {
        numerator = (int)(Math.Pow(k, n - 2) - k);
        hcf = __gcd(n, k);
    }
      
    // Condition for D-Number
    if (hcf == 1 && (numerator % n) != 0)
        return false;
      
    return true;
}
  
static int __gcd(int a, int b) 
    return b == 0 ? a : __gcd(b, a % b);     
  
// Driver Code
public static void Main(String[] args) 
{
    int n = 15;
    bool a = isDNum(n);
      
    if (a)
        Console.Write("Yes");
    else
        Console.Write("No");
}
}
  
// This code contributed by Princi Singh

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Output:

Yes

Time Complexity: O(1)
Reference: OEIS

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