在模 N 下具有乘法倒数的最接近 N 的较小数字等于该数字

给定一个素数N ，任务是找到最接近的小于N 的数，使得模 N 下的数的模乘逆等于该数本身。

例子：

Input: N = 7
Output: 6
Explanation:
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 = 11
Output: 10

朴素的方法：解决这个问题的最简单的方法是遍历从 1 到 N 的所有自然数，并找到最大的数，使得模 N 下的数的模乘法逆等于该数本身。

时间复杂度： O(N * log N)
辅助空间： O(1)

高效的方法：为了优化上述方法，该想法基于以下观察：

The nearest smaller number to N having modulo multiplicative inverse equal 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.

因此，要解决问题，只需打印N – 1作为所需答案。

下面是上述方法的实现：

C++

// C++ program to implement
// the above approach
 
#include 
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);
}

Java

// 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

Python3

# 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

C#

// 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

Javascript

// Javascript program to implement
// the above approach
 
// Function to find the nearest
// smaller number satisfying
// the condition
function clstNum(N)
{
    return (N - 1);
}
 
// Driver Code
let N = 11;
document.write(clstNum(N));
 
// This code is contributed by subham348.

输出：

10

时间复杂度： O(1)
辅助空间： O(1)