Construct Array with elements in given range and distinct GCD of each element with its index

Given 3 integers N, L and R. The task is to construct an array A[] of N integers, such that :

• Each element of the array is in the range [L, R].
• GCD(i, A[i]) are distinct for all elements.

Examples :

Input : N = 5, L = 1, R = 5
Output : {1, 2, 3, 4, 5}
Explanation : It can be seen that each element is in the range [1, 5]. 
Also, for i = 1, GCD(1, 1)=1, for i = 2, GCD(2, 2) = 2, for i = 3,  
GCD(3, 3) = 3, for i = 4, GCD(4, 4) = 4 and for i = 5, GCD(5, 5) = 5. 
Hence, all of these are distinct.

Input : N = 10, L = 30, R = 35
Output : -1
Explanation : It is not possible to construct an array 
satisfying the given conditions.

Approach: The approach of the problem is based on the following observation

To satisfy the given conditions, we will have to assure GCD(i, A[i]) = i, for each index of the array from 1 to N.

The idea is to find the  smallest possible element with gcd(i, A[i]) = i, larger than or equal to L for each i, and if that element is smaller than equal to R, then we append it, otherwise we return -1(means not possible).

 The problem can be solved by the following approach:

• Iterate from i = 1 to N.
  • For each i, find the minimum multiple of i that is strictly greater than L − 1.
  • Check if that multiple is less than or equal to R.
  • If so, that multiple would be the i^th element of the array.
• Else, array construction would not be possible.

Below is the implementation of the above approach:

// C++ code to implement the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to Construct an array whose elements
// are in given range and GCD of each element
// with its index is distinct
void constructArray(int N, int L, int R)
{
    // Declaring the array
    int A[N + 1];
    bool flag = true;

    // Constructing each element of array
    for (int i = 1; i <= N; i++) {

        // Creating ith element of the array
        // as the smallest multiple of i
        // which is greater than L-1
        A[i] = ((L - 1) / i + 1) * i;

        // Checking if the ith created element
        // is less than or equal to R
        if (A[i] > R) {
            flag = false;
            break;
        }
    }

    // If flag is false, that implies
    // it is not possible to construct the array
    if (flag == false) {
        cout << -1;
    }

    // Else print the constructed array
    else {
        for (int i = 1; i <= N; i++) {
            cout << A[i] << " ";
        }
    }
}

// Driver Code
int main()
{
    int N = 5, L = 1, R = 5;

    // Function call
    constructArray(N, L, R);
    return 0;
}

// Java code to implement the above approach
import java.io.*;

class GFG {
  // Function to Construct an array whose elements
  // are in given range and GCD of each element
  // with its index is distinct
  public static void constructArray(int N, int L, int R)
  {
    // Declaring the array
    int A[] = new int[N + 1];
    boolean flag = true;

    // Constructing each element of array
    for (int i = 1; i <= N; i++) {

      // Creating ith element of the array
      // as the smallest multiple of i
      // which is greater than L-1
      A[i] = ((L - 1) / i + 1) * i;

      // Checking if the ith created element
      // is less than or equal to R
      if (A[i] > R) {
        flag = false;
        break;
      }
    }

    // If flag is false, that implies
    // it is not possible to construct the array
    if (flag == false) {
      System.out.print(-1);
    }

    // Else print the constructed array
    else {
      for (int i = 1; i <= N; i++) {
        System.out.print(A[i] + " ");
      }
    }
  }

  // Driver Code
  public static void main(String[] args)
  {
    int N = 5, L = 1, R = 5;

    // Function call
    constructArray(N, L, R);
  }
}

// This code is contributed by Rohit Pradhan

# Python3 code to implement the above approach

# Function to Construct an array whose elements
# are in given range and GCD of each element
# with its index is distinct
def constructArray(N, L, R) :

    # Declaring the array
    A = [0] * (N + 1);
    flag = True;

    # Constructing each element of array
    for i in range(1, N + 1) :

        # Creating ith element of the array
        # as the smallest multiple of i
        # which is greater than L-1
        A[i] = ((L - 1) // i + 1) * i;

        # Checking if the ith created element
        # is less than or equal to R
        if (A[i] > R) :
            flag = False;
            break;

    # If flag is false, that implies
    # it is not possible to construct the array
    if (flag == False) :
        print(-1);

    # Else print the constructed array
    else :
        for i in range(1, N + 1) :
            print(A[i], end=" ");

# Driver Code
if __name__ == "__main__" :

    N = 5; L = 1; R = 5;

    # Function call
    constructArray(N, L, R);
   
   # This code is contributed by AnkThon

// C# code to implement the above approach
using System;

class GFG
{
  
  // Function to Construct an array whose elements
  // are in given range and GCD of each element
  // with its index is distinct
  public static void constructArray(int N, int L, int R)
  {
    
    // Declaring the array
    int []A = new int[N + 1];
    bool flag = true;

    // Constructing each element of array
    for (int i = 1; i <= N; i++) {

      // Creating ith element of the array
      // as the smallest multiple of i
      // which is greater than L-1
      A[i] = ((L - 1) / i + 1) * i;

      // Checking if the ith created element
      // is less than or equal to R
      if (A[i] > R) {
        flag = false;
        break;
      }
    }

    // If flag is false, that implies
    // it is not possible to construct the array
    if (flag == false) {
      Console.Write(-1);
    }

    // Else print the constructed array
    else {
      for (int i = 1; i <= N; i++) {
        Console.Write(A[i] + " ");
      }
    }
  }

  // Driver Code
  public static void Main(string[] args)
  {
    int N = 5, L = 1, R = 5;

    // Function call
    constructArray(N, L, R);
  }
}

// This code is contributed by AnkThon

<script>
// Javascript code to implement the above approach

// Function to Construct an array whose elements
// are in given range and GCD of each element
// with its index is distinct
function constructArray(N, L, R)
{
    // Declaring the array
   let A= new Array(N + 1);
    let flag = true;

    // Constructing each element of array
    for (let i = 1; i <= N; i++) {

        // Creating ith element of the array
        // as the smallest multiple of i
        // which is greater than L-1
        A[i] = ((L - 1) / i + 1) * i;

        // Checking if the ith created element
        // is less than or equal to R
        if (A[i] > R) {
            flag = false;
            break;
        }
    }

    // If flag is false, that implies
    // it is not possible to construct the array
    if (flag == false) {
        document.write(-1);
    }

    // Else print the constructed array
    else {
        for (let i = 1; i <= N; i++) {
            document.write(A[i] + " ");
        }
    }
}

// Driver Code

    let N = 5, L = 1, R = 5;

    // Function call
    constructArray(N, L, R);
    
    // This code is contributed by satwik4409.
    </script>

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