Given a prime number **N**, the task is to find the closest smaller number than **N** such that modulo multiplicative inverse of a number under modulo N is equal to the number itself.

**Examples:**

Input:N = 7Output:6Explanation:

Modulo multiplicative inverse of all possible natural numbers from 1 to less than N are:

Modulo multiplicative inverse of 1 under modulo N(=7) is 1.

Modulo multiplicative inverse of 2 under modulo N(=7) is 4.

Modulo multiplicative inverse of 3 under modulo N(=7) is 5.

Modulo multiplicative inverse of 4 under modulo N(=7) is 2.

Modulo multiplicative inverse of 5 under modulo N(=7) is 3.

Modulo multiplicative inverse of 6 under modulo N(=7) is 6.

Therefore, the nearest smaller number to N(= 7) having modulo inverse equal to the number itself is 6.

Input: N= 11Output:10

**Naive Approach: **The simplest approach to solve this problem is to traverse all natural numbers from 1 to N and find the largest number such that modulo multiplicative inverse of the number under modulo N is equal to the number itself.

**Time Complexity:** O(N * log N)**Auxiliary Space:** O(1)

**Efficient Approach: **To optimize the above approach the idea is based on the following observations:

The nearest smaller number to

Nhavingmodulo multiplicative inverseequal to the number itself is(N – 1).

Mathematical proof:

If X and Y are two numbers such that (X * Y) % N = 1 mod(N), then Y is modulo inverse of X.

Put X = N – 1 then

=>((N – 1) * Y) % N = 1 mod(N)

=>(N × Y) % N – Y % N = 1 mod(N)

=> Y = N – 1

Therefore, for X = N – 1 the value of Y is equal to X.

Therefore, to solve the problem, simply print **N – 1** as the required answer.

Below is the implementation of the above approach:

`// C++ program to implement` `// the above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;`
`// Function to find the nearest` `// smaller number satisfying` `// the condition` `int` `clstNum(` `int` `N)`
`{` ` ` `return` `(N - 1);`
`}` `// Driver Code` `int` `main()`
`{` ` ` `int` `N = 11;`
` ` `cout << clstNum(N);`
`}` |

*chevron_right*

*filter_none*

`// Java program to implement` `// the above approach` `import` `java.io.*;`
`class` `GFG{`
`// Function to find the nearest` `// smaller number satisfying` `// the condition` `static` `int` `clstNum(` `int` `N){ ` `return` `(N - ` `1` `); }`
`// Driver Code` `public` `static` `void` `main(String[] args)`
`{` ` ` `int` `N = ` `11` `;`
` ` ` ` `System.out.println(clstNum(N));`
`}` `}` `// This code is contributed by akhilsaini` |

*chevron_right*

*filter_none*

`# Python3 program to implement` `# the above approach` `# Function to find the nearest` `# smaller number satisfying` `# the condition` `def` `clstNum(N):`
` ` `return` `(N ` `-` `1` `)`
`# Driver Code` `if` `__name__ ` `=` `=` `'__main__'` `:`
` ` ` ` `N ` `=` `11`
` ` ` ` `print` `(clstNum(N))`
` ` `# This code is contributed by akhilsaini` |

*chevron_right*

*filter_none*

`// C# program to implement` `// the above approach` `using` `System;`
`class` `GFG{`
`// Function to find the nearest` `// smaller number satisfying` `// the condition` `static` `int` `clstNum(` `int` `N){ ` `return` `(N - 1); }`
`// Driver Code` `public` `static` `void` `Main()`
`{` ` ` `int` `N = 11;`
` ` ` ` `Console.Write(clstNum(N));`
`}` `}` `// This code is contributed by akhilsaini` |

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*filter_none*

**Output:**

10

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

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