Generate an array with product of all subarrays of length exceeding one divisible by K

Given two positive integers N and K, the task is to generate an array of length N such that the product of every subarray of length greater than 1 must be divisible by K and the maximum element of the array must be less than K. If no such array is possible, then print -1.

Examples:

Input: N = 3, K = 20
Output: {15, 12, 5}
Explanation: All subarrays of length greater than 1 are {15, 12}, {12, 5}, {15, 12, 5} and their corresponding products are 180, 60 and 900, which are all divisible by 20.

Input: N = 4, K = 100
Output: {90, 90, 90, 90}

Approach: The given problem can be solved by the following observations:

It can be observed that by making the product of every subarray of length 2 divisible by K, the product of every subarray of length greater than 2 will automatically be divisible by K.

Therefore, the idea is to take two divisors of K, say d1 and d2, such that d1 * d2 = K and place them alternatively in the array. Follow the steps below to solve the problem:

1. Initialize two integer variables d1 and d2.
2. Check if K is prime. If found to be true, print -1.
3. Otherwise, calculate the divisors of K and store two divisors in d1 and d2.
4. After that, traverse from i = 0 to N - 1.
5. Print d1 if i is even. Otherwise, print d2.

Below is the implementation of the above approach:

// C++ program for the above approach

#include <iostream>
using namespace std;

// Function to check if the required
// array can be generated or not
void array_divisbleby_k(int N, int K)
{
    // To check if divisor exists
    bool flag = false;

    // To store divisors of K
    int d1, d2;

    // Check if K is prime or not
    for (int i = 2; i * i <= K; i++) {

        if (K % i == 0) {
            flag = true;
            d1 = i;
            d2 = K / i;
            break;
        }
    }

    // If array can be generated
    if (flag) {

        // Print d1 and d2 alternatively
        for (int i = 0; i < N; i++) {

            if (i % 2 == 1) {
                cout << d2 << " ";
            }
            else {
                cout << d1 << " ";
            }
        }
    }

    else {

        // No such array can be generated
        cout << -1;
    }
}
// Driver Code
int main()
{
    // Given N and K
    int N = 5, K = 21;

    // Function Call
    array_divisbleby_k(N, K);

    return 0;
}

// Java program for the above approach
class GFG{
    
// Function to check if the required 
// array can be generated or not 
public static void array_divisbleby_k(int N, 
                                      int K) 
{ 
    
    // To check if divisor exists 
    boolean flag = false; 
  
    // To store divisors of K 
    int d1 = 0, d2 = 0; 
  
    // Check if K is prime or not 
    for(int i = 2; i * i <= K; i++)
    { 
        if (K % i == 0)
        {
            flag = true; 
            d1 = i; 
            d2 = K / i; 
            break; 
        } 
    } 
  
    // If array can be generated 
    if (flag) 
    { 
        
        // Print d1 and d2 alternatively 
        for(int i = 0; i < N; i++) 
        { 
            if (i % 2 == 1) 
            { 
                System.out.print(d2 + " ");
            } 
            else
            { 
                System.out.print(d1 + " ");
            } 
        } 
    } 
  
    else 
    { 
        
        // No such array can be generated 
        System.out.print(-1);
    } 
} 

// Driver Code
public static void main(String[] args) 
{
    
    // Given N and K 
    int N = 5, K = 21; 
  
    // Function Call 
    array_divisbleby_k(N, K); 
}
}

// This code is contributed by divyesh072019

# Python3 program for the above approach

# Function to check if the required
# array can be generated or not
def array_divisbleby_k(N, K):

    # To check if divisor exists
    flag = False

    # To store divisors of K
    d1, d2 = 0, 0

    # Check if K is prime or not
    for i in range(2, int(K ** (1 / 2)) + 1):
        if (K % i == 0):
            flag = True
            d1 = i
            d2 = K // i
            break

    # If array can be generated
    if (flag):

        # Print d1 and d2 alternatively
        for i in range(N):
            if (i % 2 == 1):
                print(d2, end = " ")
            else:
                print(d1, end = " ")
                
    else:

        # No such array can be generated
        print(-1)
 
# Driver Code
if __name__ == "__main__":

    # Given N and K
    N = 5
    K = 21

    # Function Call
    array_divisbleby_k(N, K)

# This code is contributed by AnkThon

// C# program for the above approach
using System;

class GFG{
    
// Function to check if the required 
// array can be generated or not 
public static void array_divisbleby_k(int N, 
                                      int K) 
{ 
    
    // To check if divisor exists 
    bool flag = false; 
    
    // To store divisors of K 
    int d1 = 0, d2 = 0; 
  
    // Check if K is prime or not 
    for(int i = 2; i * i <= K; i++)
    { 
        if (K % i == 0)
        {
            flag = true; 
            d1 = i; 
            d2 = K / i; 
            break; 
        } 
    } 
  
    // If array can be generated 
    if (flag) 
    { 
        
        // Print d1 and d2 alternatively 
        for(int i = 0; i < N; i++) 
        { 
            if (i % 2 == 1) 
            { 
                Console.Write(d2 + " ");
            } 
            else
            { 
                Console.Write(d1 + " ");
            } 
        } 
    } 
  
    else 
    { 
        
        // No such array can be generated 
        Console.Write(-1);
    } 
} 

// Driver Code
public static void Main(string[] args) 
{
    
    // Given N and K 
    int N = 5, K = 21; 
  
    // Function Call 
    array_divisbleby_k(N, K); 
}
}

// This code is contributed by AnkThon

<script>
    // Javascript program for the above approach
    
    // Function to check if the required
    // array can be generated or not
    function array_divisbleby_k(N, K)
    {

        // To check if divisor exists
        let flag = false;

        // To store divisors of K
        let d1 = 0, d2 = 0;

        // Check if K is prime or not
        for(let i = 2; i * i <= K; i++)
        {
            if (K % i == 0)
            {
                flag = true;
                d1 = i;
                d2 = K / i;
                break;
            }
        }

        // If array can be generated
        if (flag)
        {

            // Print d1 and d2 alternatively
            for(let i = 0; i < N; i++)
            {
                if (i % 2 == 1)
                {
                    document.write(d2 + " ");
                }
                else
                {
                    document.write(d1 + " ");
                }
            }
        }

        else
        {

            // No such array can be generated
            document.write(-1);
        }
    }
    
    // Given N and K
    let N = 5, K = 21;
   
    // Function Call
    array_divisbleby_k(N, K);

// This code is contributed by suresh07.
</script>

Output:

3 7 3 7 3

Time complexity: O(N + ?K)
Auxiliary space: O(1)