Largest subsequence having GCD greater than 1

Given an array arr[], find the largest subsequence such that GCD of all those subsequence are greater than 1.
Examples:

Input: 3, 6, 2, 5, 4
Output: 3
Explanation: There are only three elements(6, 
2, 4) having GCD greater than 1 i.e., 2. So the 
largest subsequence will be 3

Input: 10, 15, 7, 25, 9, 35
Output: 4

Naive Approach(Method 1)

Simple approach is to generate all the subsequence one by one and then find the GCD of all such generated set. Problem of this approach is that it grows exponentially in 2N



Iterative Approach(Method 2)

If we observe then we will found that to make gcd greater than 1, all such elements must contain comman factor greater than 1 which evenly divides all these values. So in order to get that factor we will iterate from 2 to Maximum element of array and then check for divisibility.

C++

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// Simple C++ program to find length of
// the largest subsequence with GCD greater
// than 1.
#include<bits/stdc++.h>
  
using namespace std;
  
// Returns length of the largest subsequence
// with GCD more than 1.
int largestGCDSubsequence(int arr[], int n)
{
    int ans = 0;
  
    // Finding the Maximum value in arr[]
    int maxele = *max_element(arr, arr+n);
  
    // Iterate from 2 to maximum possible
    // divisor of all give values
    for (int i=2; i<=maxele; ++i)
    {
        int count = 0;
        for (int j=0; j<n; ++j)
        {
            // If we found divisor,
            // increment count
            if (arr[j]%i == 0)
                ++count;
        }
        ans = max(ans, count);
    }
  
    return ans;
}
  
// Driver code
int main()
{
    int arr[] = {3, 6, 2, 5, 4};
    int size = sizeof(arr) / sizeof(arr[0]);
    cout << largestGCDSubsequence(arr, size);
    return 0;
}

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Java

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// Efficient Java program to find length of
// the largest subsequence with GCD greater
// than 1.
import java.util.Arrays;
  
class GFG {
// Returns length of the largest subsequence
// with GCD more than 1.
static int largestGCDSubsequence(int arr[], int n)
{
    int ans = 0;
   
    // Finding the Maximum value in arr[]
    int maxele = Arrays.stream(arr).max().getAsInt();;
   
    // Iterate from 2 to maximum possible
    // divisor of all give values
    for (int i=2; i<=maxele; ++i)
    {
        int count = 0;
        for (int j=0; j<n; ++j)
        {
            // If we found divisor,
            // increment count
            if (arr[j]%i == 0)
                ++count;
        }
        ans = Math.max(ans, count);
    }
   
    return ans;
}
// Driver program to test above
  
    public static void main(String[] args) {
  
        int arr[] = {3, 6, 2, 5, 4};
        int size = arr.length;
  
        System.out.println(largestGCDSubsequence(arr, size));
    }
}
  
//this code contributed by Rajput-Ji

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Python3

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# Simple Python 3 program to find length of
# the largest subsequence with GCD greater
# than 1.
  
# Returns length of the largest subsequence
# with GCD more than 1.
def largestGCDSubsequence(arr, n):
    ans = 0
  
    # Finding the Maximum value in arr[]
    maxele = max(arr)
  
    # Iterate from 2 to maximum possible
    # divisor of all give values
    for i in range(2, maxele + 1):
        count = 0
        for j in range(n):
              
            # If we found divisor,
            # increment count
            if (arr[j] % i == 0):
                count += 1
        ans = max(ans, count)
  
    return ans
  
# Driver code
if __name__ == '__main__':
    arr = [3, 6, 2, 5, 4]
    size = len(arr)
    print(largestGCDSubsequence(arr, size))
  
# This code is contributed by Rajput-Ji

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C#

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// Efficient C# program to find length of
// the largest subsequence with GCD greater
// than 1.
using System;
using System.Linq;
public class GFG {
// Returns length of the largest subsequence
// with GCD more than 1.
static int largestGCDSubsequence(int []arr, int n)
{
    int ans = 0;
    
    // Finding the Maximum value in arr[]
    int maxele = arr.Max();
    
    // Iterate from 2 to maximum possible
    // divisor of all give values
    for (int i=2; i<=maxele; ++i)
    {
        int count = 0;
        for (int j=0; j<n; ++j)
        {
            // If we found divisor,
            // increment count
            if (arr[j]%i == 0)
                ++count;
        }
        ans = Math.Max(ans, count);
    }
    
    return ans;
}
// Driver program to test above
   
    public static void Main() {
   
        int []arr = {3, 6, 2, 5, 4};
        int size = arr.Length;
   
        Console.Write(largestGCDSubsequence(arr, size));
    }
}
   
//this code contributed by Rajput-Ji

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PHP

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<?php
// Simple PHP program to find length of 
// the largest subsequence with GCD greater 
// than 1. 
  
// Returns length of the largest subsequence 
// with GCD more than 1. 
function largestGCDSubsequence($arr, $n
    $ans = 0; 
  
    // Finding the Maximum value in arr[] 
    $maxele = max($arr); 
  
    // Iterate from 2 to maximum possible 
    // divisor of all give values 
    for ($i = 2; $i <= $maxele; ++$i
    
        $count = 0; 
        for ($j = 0; $j < $n; ++$j
        
            // If we found divisor, 
            // increment count 
            if ($arr[$j] % $i == 0) 
                ++$count
        
        $ans = max($ans, $count); 
    
  
    return $ans
  
// Driver code 
$arr = array(3, 6, 2, 5, 4); 
$size = count($arr); 
echo largestGCDSubsequence($arr, $size);
      
// This code is contributed by mits
?>

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Output:

3

Time Complexity: O(n * max(arr[i])) where n is size of array.
Auxiliary Space: O(1)

Best Approach(Method 3)

An efficient approach is to use prime factorization method with the help of Sieve of Eratosthenes. First of all we will find the smallest prime divisor of all elements by pre-computed sieve. After that we will mark all the prime divisor of every element of arr[] by factorizing it with the help of pre-computed prime[] array.
Now we have all the marked primes occurring in all the array elements. The last step is to find the maximum count of all such prime factors.

C++

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// Efficient C++ program to find length of
// the largest subsequence with GCD greater
// than 1.
#include<bits/stdc++.h>
  
using namespace std;
  
#define MAX 100001
  
// prime[] for storing smallest prime divisor of element
// count[] for storing the number of times a particular
// divisor occurs in a subsequence
int prime[MAX], countdiv[MAX];
  
// Simple sieve to find smallest prime factors of numbers
// smaller than MAX
void SieveOfEratosthenes()
{
    for (int i = 2; i * i <= MAX; ++i)
    {
        if (!prime[i])
            for (int j = i * 2; j <= MAX; j += i)
                prime[j] = i;
    }
  
    // Prime number will have same divisor
    for (int i = 1; i < MAX; ++i)
        if (!prime[i])
            prime[i] = i;
}
  
// Returns length of the largest subsequence
// with GCD more than 1.
int largestGCDSubsequence(int arr[], int n)
{
    int ans = 0;
    for (int i=0; i < n; ++i)
    {
        int element = arr[i];
  
        // Fetch total unique prime divisor of element
        while (element > 1)
        {
            int div = prime[element];
  
            // Increment count[] of Every unique divisor
            // we get till now
            ++countdiv[div];
  
            // Find maximum frequency of divisor
            ans = max(ans, countdiv[div]);
  
            while (element % div==0)
                element /= div;
        }
    }
  
    return ans;
}
  
// Driver code
int main()
{
    // Pre-compute smallest divisor of all numbers
    SieveOfEratosthenes();
  
    int arr[] = {10, 15, 7, 25, 9, 35};
    int size = sizeof(arr) / sizeof(arr[0]);
  
    cout << largestGCDSubsequence(arr, size);
    return 0;
}

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Java

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// Efficient Java program to find length of
// the largest subsequence with GCD greater
// than 1.
  
class GFG
{
static int MAX = 100001;
  
// prime[] for storing smallest prime divisor
// of element count[] for storing the number
// of times a particular divisor occurs 
// in a subsequence
static int[] prime = new int[MAX + 1];
static int[] countdiv = new int[MAX + 1];
  
// Simple sieve to find smallest prime 
// factors of numbers smaller than MAX
static void SieveOfEratosthenes()
{
    for (int i = 2; i * i <= MAX; ++i)
    {
        if (prime[i] == 0)
            for (int j = i * 2; j <= MAX; j += i)
                prime[j] = i;
    }
  
    // Prime number will have same divisor
    for (int i = 1; i < MAX; ++i)
        if (prime[i] == 0)
            prime[i] = i;
}
  
// Returns length of the largest subsequence
// with GCD more than 1.
static int largestGCDSubsequence(int arr[], int n)
{
    int ans = 0;
    for (int i = 0; i < n; ++i)
    {
        int element = arr[i];
  
        // Fetch total unique prime divisor of element
        while (element > 1)
        {
            int div = prime[element];
  
            // Increment count[] of Every unique divisor
            // we get till now
            ++countdiv[div];
  
            // Find maximum frequency of divisor
            ans = Math.max(ans, countdiv[div]);
  
            while (element % div == 0)
                element /= div;
        }
    }
    return ans;
}
  
// Driver code
public static void main (String[] args)
{
    // Pre-compute smallest divisor of all numbers
    SieveOfEratosthenes();
  
    int arr[] = {10, 15, 7, 25, 9, 35};
    int size = arr.length;
  
    System.out.println(largestGCDSubsequence(arr, size));
}
}
  
// This code is contributed by mits

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Python3

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# Efficient Python3 program to find length 
# of the largest subsequence with GCD 
# greater than 1.
import math as mt
  
MAX = 100001
  
# prime[] for storing smallest
# prime divisor of element
# count[] for storing the number
# of times a particular divisor 
# occurs in a subsequence
prime = [0 for i in range(MAX + 1)]
countdiv = [0 for i in range(MAX + 1)]
  
# Simple sieve to find smallest prime 
# factors of numbers smaller than MAX
def SieveOfEratosthenes():
  
    for i in range(2, mt.ceil(mt.sqrt(MAX + 1))):
      
        if (prime[i] == 0):
            for j in range(i * 2, MAX + 1, i):
                prime[j] = i
      
    # Prime number will have same divisor
    for i in range(1, MAX):
        if (prime[i] == 0):
            prime[i] = i
  
# Returns length of the largest 
# subsequence with GCD more than 1.
def largestGCDSubsequence(arr, n):
  
    ans = 0
    for i in range(n):
  
        element = arr[i]
  
        # Fetch total unique prime
        # divisor of element
        while (element > 1):
  
            div = prime[element]
  
            # Increment count[] of Every 
            # unique divisor we get till now
            countdiv[div] += 1
  
            # Find maximum frequency of divisor
            ans = max(ans, countdiv[div])
  
            while (element % div == 0):
                element = element // div
          
    return ans
  
# Driver code
  
# Pre-compute smallest divisor
# of all numbers
SieveOfEratosthenes()
  
arr= [10, 15, 7, 25, 9, 35]
size = len(arr) 
print(largestGCDSubsequence(arr, size))
  
# This code is contributed 
# by Mohit kumar 29

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C#

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// Efficient C# program to find length of 
// the largest subsequence with GCD greater 
// than 1. 
using System;
  
class GFG
{
      
static int MAX=100001; 
  
// prime[] for storing smallest 
// prime divisor of element count[]
// for storing the number of times 
// a particular divisor occurs in a subsequence 
static int[] prime = new int[MAX + 1];
static int[] countdiv = new int[MAX + 1]; 
  
// Simple sieve to find smallest prime 
//  factors of numbers smaller than MAX 
static void SieveOfEratosthenes() 
    for (int i = 2; i * i <= MAX; ++i) 
    
        if (prime[i] == 0) 
            for (int j = i * 2; j <= MAX; j += i) 
                prime[j] = i; 
    
  
    // Prime number will have same divisor 
    for (int i = 1; i < MAX; ++i) 
        if (prime[i] == 0) 
            prime[i] = i; 
  
// Returns length of the largest subsequence 
// with GCD more than 1. 
static int largestGCDSubsequence(int []arr, int n) 
    int ans = 0; 
    for (int i = 0; i < n; ++i) 
    
        int element = arr[i]; 
  
        // Fetch total unique prime divisor of element 
        while (element > 1) 
        
            int div = prime[element]; 
  
            // Increment count[] of Every unique divisor 
            // we get till now 
            ++countdiv[div]; 
  
            // Find maximum frequency of divisor 
            ans = Math.Max(ans, countdiv[div]); 
  
            while (element % div==0) 
                element /= div; 
        
    
    return ans; 
  
// Driver code 
public static void Main() 
{
    // Pre-compute smallest 
    // divisor of all numbers 
    SieveOfEratosthenes(); 
  
    int []arr = {10, 15, 7, 25, 9, 35}; 
    int size = arr.Length; 
  
    Console.WriteLine(largestGCDSubsequence(arr, size)); 
}
  
// This code is contributed by mits

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PHP

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<?php
// Efficient PHP program to find length of
// the largest subsequence with GCD greater
// than 1.
  
$MAX = 10001;
  
// prime[] for storing smallest prime divisor of element
// count[] for storing the number of times a particular
// divisor occurs in a subsequence
$prime = array_fill(0, $MAX, 0);
$countdiv = array_fill(0, $MAX, 0);
  
// Simple sieve to find smallest prime factors of numbers
// smaller than MAX
function SieveOfEratosthenes()
{
    global $MAX,$prime;
    for ($i = 2; $i * $i <= $MAX; ++$i)
    {
        if ($prime[$i] == 0)
            for ($j = $i * 2; $j <= $MAX; $j += $i)
                $prime[$j] = $i;
    }
  
    // Prime number will have same divisor
    for ($i = 1; $i < $MAX; ++$i)
        if ($prime[$i] == 0)
            $prime[$i] = $i;
}
  
// Returns length of the largest subsequence
// with GCD more than 1.
function largestGCDSubsequence($arr, $n)
{
    global $countdiv,$prime;
    $ans = 0;
    for ($i = 0; $i < $n; ++$i)
    {
        $element = $arr[$i];
  
        // Fetch total unique prime divisor of element
        while ($element > 1)
        {
            $div = $prime[$element];
  
            // Increment count[] of Every unique divisor
            // we get till now
            ++$countdiv[$div];
  
            // Find maximum frequency of divisor
            $ans = max($ans, $countdiv[$div]);
  
            while ($element % $div == 0)
                $element = (int)($element/$div);
        }
    }
  
    return $ans;
}
  
    // Driver code
    // Pre-compute smallest divisor of all numbers
    SieveOfEratosthenes();
  
    $arr = array(10, 15, 7, 25, 9, 35);
    $size = count($arr);
  
    echo largestGCDSubsequence($arr, $size);
  
// This code is contributed by mits
?>

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Output:

 4

Time complexity: O( n*log(max(arr[i])) ) + MAX*log(log(MAX))
Auxiliary space: O(MAX)

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