# 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

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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++

 `// C++ implementation  ` `// for the above approach ` `#include ` `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 `

## Java

 `// 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 `

## Python3

 `# 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"``) `

## C#

 `// 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 `

Output:

```Yes
```

Time Complexity: O(1)
Reference: OEIS

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