Check if an array can be split into subarrays with GCD exceeding K

Given an array arr[] of N integers and a positive integer K, the task is to check if it is possible to split this array into distinct contiguous subarrays such that the Greatest Common Divisor of all elements of each subarray is greater than K.

Note: Each array element can be a part of exactly one subarray.

Examples:

Input: arr[] = {3, 2, 4, 4, 8}, K = 1
Output: Yes
Explanation:
One valid split is [3], [2, 4], [4, 8] with GCD 3, 2 and 4 respectively. 
Another Valid Split is [3], [2, 4], [4], [8] with GCD 3, 2, 4 and 8 respectively.
Therefore, the given array can be split into subarrays having GCD > K.

Input: arr[] = {2, 4, 6, 1, 8, 16}, K = 3
Output: No

Approach: This problem can be solved using the following observations:



  • If any array element is found to be less than or equal to K, then the answer is always “No”. This is because the subarray that contains this number will always have GCD less than or equal to K.
  • If no array element is found to be less than or equal to K, then it is always possible to divide the entire array into N subarrays each of size at least 1 whose GCD is always greater than K.

Therefore, from the above observations, the idea is to traverse the given array and check that if there exists any element in the array which is less than or equal to K. If found to be true, then print “No”. Otherwise print “Yes”.

Below is the implementation of the above approach:

C++

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// C++ program for the above approach
  
#include <iostream>
using namespace std;
  
// Function to check if it is possible
// to split an array into subarrays
// having GCD at least K
string canSplitArray(int arr[], int n,
                     int k)
{
  
    // Iterate over the array arr[]
    for (int i = 0; i < n; i++) {
  
        // If the current element
        // is less than or equal to k
        if (arr[i] <= k) {
            return "No";
        }
    }
  
    // If no array element is found
    // to be less than or equal to k
    return "Yes";
}
  
// Driver Code
int main()
{
    // Given array arr[]
    int arr[] = { 2, 4, 6, 1, 8, 16 };
  
    int N = sizeof arr / sizeof arr[0];
  
    // Given K
    int K = 3;
  
    // Function Call
    cout << canSplitArray(arr, N, K);
}

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Java

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// Java program for the above approach
import java.io.*; 
  
class GFG{ 
      
// Function to check if it is possible
// to split an array into subarrays
// having GCD at least K
static String canSplitArray(int arr[], 
                            int n, int k)
{
      
    // Iterate over the array arr[]
    for(int i = 0; i < n; i++) 
    {
          
        // If the current element
        // is less than or equal to k
        if (arr[i] <= k) 
        {
            return "No";
        }
    }
      
    // If no array element is found
    // to be less than or equal to k
    return "Yes";
}
  
// Driver code 
public static void main (String[] args) 
      
    // Given array arr[]
    int arr[] = { 2, 4, 6, 1, 8, 16 };
      
    // Length of the array 
    int N = arr.length;
      
    // Given K
    int K = 3;
      
    // Function call 
    System.out.println(canSplitArray(arr, N, K));
  
// This code is contributed by jana_sayantan

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Python3

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# Python3 program for the above approach
  
# Function to check if it is possible
# to split an array into subarrays
# having GCD at least K
def canSplitArray(arr, n, k):
  
    # Iterate over the array arr[]
    for i in range(n):
  
        # If the current element
        # is less than or equal to k
        if (arr[i] <= k):
            return "No"
  
    # If no array element is found
    # to be less than or equal to k
    return "Yes"
  
# Driver Code
if __name__ == '__main__':
  
    # Given array arr[]
    arr = [ 2, 4, 6, 1, 8, 16 ]
  
    N = len(arr)
  
    # Given K
    K = 3
  
    # Function call
    print(canSplitArray(arr, N, K))
  
# This code is contributed by mohit kumar 29

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C#

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// C# program for the above approach
using System;
  
class GFG{ 
      
// Function to check if it is possible
// to split an array into subarrays
// having GCD at least K
static String canSplitArray(int []arr, 
                            int n, int k)
{
      
    // Iterate over the array []arr
    for(int i = 0; i < n; i++) 
    {
          
        // If the current element
        // is less than or equal to k
        if (arr[i] <= k) 
        {
            return "No";
        }
    }
      
    // If no array element is found
    // to be less than or equal to k
    return "Yes";
}
  
// Driver code 
public static void Main(String[] args) 
      
    // Given array []arr
    int []arr = { 2, 4, 6, 1, 8, 16 };
      
    // Length of the array 
    int N = arr.Length;
      
    // Given K
    int K = 3;
      
    // Function call 
    Console.WriteLine(canSplitArray(arr, N, K));
  
// This code is contributed by Amit Katiyar

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Output: 

No

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

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