Number of subarrays with GCD equal to 1 in O(n) time

Given an array arr[] of size N, the task is to find the maximum number of sub-arrays such that the GCD(Greatest Common Factor) of all elements of each sub-array is equal to 1 in O(n).

Examples:

Input: arr[] = {4, 2, 3, 0, 1, 8, 16, 1}.
Output: 3
Explanation: GCD of subarray {4, 2, 3} is 1, GCD of subarray {0, 1} is 1. GCD of {8, 16, 1} is 1, maximum number of subarrays is 3.

Input: arr[] = {9, 36, 8, 3, 7, 2, 13, 26, 5}
Output: 4
Explanation: GCD of subarray {9, 36, 8} is 1, GCD of subarray {3, 7} is 1, GCD of {2, 13} is 1, GCD of {26, 5} is 1, maximum number of subarrays is 4.

Approach: To solve the problem follow the below observations:

• GCD of any number with 0 is the number itself, hence, initially set GCD as 0.
• GCD of any two numbers will be 1 if they don't have any common factor.

In the traversal:

• Keep on calculating the GCD of the current element with the GCD of the subarray till now.
• At any position, If the GCD of the current subarray becomes 1 then increase the count of the subarray by one and set the gcd to 0 so that we can calculate the maximum subarray.
• Return the number of subarrays.

Follow the given steps to solve the problem:

• Declare variables cnt and g and initialize them to 0.
• variable g denotes GCD.
• While traversing the array:
  • Keep calculating GCD till the g becomes 1.
  • If g equals 1, then increase the cnt variable, and set g to 0.
• In the end, return the number of subarrays.

Below is the implementation for the above approach:

// C++ program to find maximum number of
// sub-arrays such that the GCD
// of all elements of each sub-array is
// equal to 1 in O(n))
#include <bits/stdc++.h>
using namespace std;

// Function to find GCD
int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}

// Function to count sub-arrays
int count_subarrays(int arr[], int n)
{
    int g = 0;
    int cnt = 0;

    // Counting subarrays whose gcd is 1
    for (int i = 0; i < n; ++i) {
        g = gcd(g, arr[i]);

        // gcd becomes 1 then increases 
        // the count
        if (g == 1) {
            cnt++;

            // Set g to 0 for further use
            g = 0;
        }
    }

    // Returning no. of subarrays
    return cnt;
}

// Driver code
int main()
{
    int arr[] = { 9, 36, 8, 3, 7, 2, 13, 26, 5 };
    int n = (sizeof(arr) / sizeof(arr[0]));

    // Function call
    int ans = count_subarrays(arr, n);
    cout << ans;
    return 0;
}

// Java program to find maximum number of
// sub-arrays such that the GCD
// of all elements of each sub-array is
// equal to 1 in O(n))
import java.io.*;

class GFG {
    // Function to find GCD
    public static int gcd(int a, int b)
    {
        if (b == 0)
            return a;
        return gcd(b, a % b);
    }

    // Function to count sub-arrays
    public static int count_subarrays(int arr[], int n)
    {
        int g = 0;
        int cnt = 0;

        // Counting subarrays whose gcd is 1
        for (int i = 0; i < n; ++i) {
            g = gcd(g, arr[i]);

            // gcd becomes 1 then increases
            // the count
            if (g == 1) {
                cnt++;

                // Set g to 0 for further use
                g = 0;
            }
        }

        // Returning no. of subarrays
        return cnt;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int arr[] = { 9, 36, 8, 3, 7, 2, 13, 26, 5 };
        int n = arr.length;

        // Function call
        int ans = count_subarrays(arr, n);
        System.out.print(ans);
    }
}

// This code is contributed by Rohit Pradhan

class GFG :
    # Function to find GCD
    @staticmethod
    def  gcd( a,  b) :
        if (b == 0) :
            return a
        return GFG.gcd(b, a % b)
      
    # Function to count sub-arrays
    @staticmethod
    def  count_subarrays( arr,  n) :
        g = 0
        cnt = 0
        
        # Counting subarrays whose gcd is 1
        i = 0
        while (i < n) :
            g = GFG.gcd(g, arr[i])
            
            # gcd becomes 1 then increases
            # the count
            if (g == 1) :
                cnt += 1
                
                # Set g to 0 for further use
                g = 0
            i += 1
            
        # Returning no. of subarrays
        return cnt
      
    # Driver Code
    @staticmethod
    def main( args) :
        arr = [9, 36, 8, 3, 7, 2, 13, 26, 5]
        n = len(arr)
        
        # Function call
        ans = GFG.count_subarrays(arr, n)
        print(ans, end ="")
    
if __name__=="__main__":
    GFG.main([])
    
    # This code is contributed by aadityaburujwale.

// C# program to find maximum number of
// sub-arrays such that the GCD
// of all elements of each sub-array is
// equal to 1 in O(n))
using System;
public class GFG
{
  
    // Function to find GCD
    public static int gcd(int a, int b)
    {
        if (b == 0)
            return a;
        return gcd(b, a % b);
    }

    // Function to count sub-arrays
    public static int count_subarrays(int []arr, int n)
    {
        int g = 0;
        int cnt = 0;

        // Counting subarrays whose gcd is 1
        for (int i = 0; i < n; ++i) {
            g = gcd(g, arr[i]);

            // gcd becomes 1 then increases
            // the count
            if (g == 1) {
                cnt++;

                // Set g to 0 for further use
                g = 0;
            }
        }

        // Returning no. of subarrays
        return cnt;
    }

    // Driver Code
    public static void Main(string[] args)
    {
        int []arr = { 9, 36, 8, 3, 7, 2, 13, 26, 5 };
        int n = arr.Length;

        // Function call
        int ans = count_subarrays(arr, n);
        Console.WriteLine(ans);
    }
}

// This code is contributed by AnkThon

// JS program to find maximum number of
// sub-arrays such that the GCD
// of all elements of each sub-array is
// equal to 1 in O(n))

// Function to find GCD
function gcd( a, b)
{
    if (b == 0)
        return a;
    return gcd(b,a % b);
}

// Function to count sub-arrays
function count_subarrays(arr,  n)
{
    let g = 0;
     let cnt = 0;

    // Counting subarrays whose gcd is 1
    for (let i = 0; i < n; ++i) {
        g = gcd(g, arr[i]);

        // gcd becomes 1 then increases 
        // the count
        if (g == 1) {
            cnt++;

            // Set g to 0 for further use
            g = 0;
        }
    }

    // Returning no. of subarrays
    return cnt;
}

// Driver code
    let arr = [ 9, 36, 8, 3, 7, 2, 13, 26, 5 ];
    let n = arr.length;

    // Function call
    let ans = count_subarrays(arr, n);
    console.log(ans);
    
//this code is contributed by ksam24000

4

Time Complexity: O(N*log(M)), where N is the size of the array and M is the minimum element in the given array
Auxiliary Space: O(1)