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Count Divisors of n in O(n^1/3)

  • Difficulty Level : Hard
  • Last Updated : 08 Jul, 2021

Given a number n, count all distinct divisors of it.

Examples: 

Input : 18
Output : 6
Divisors of 18 are 1, 2, 3, 6, 9 and 18.

Input : 100
Output : 9
Divisors of 100 are 1, 2, 4, 5, 10, 20,
25, 50 and 100

A Naïve Solution would be to iterate all the numbers from 1 to sqrt(n), checking if that number divides n and incrementing number of divisors. This approach takes O(sqrt(n)) time. 

C++




// C implementation of Naive method to count all
// divisors
#include <bits/stdc++.h>
using namespace std;
 
// function to count the divisors
int countDivisors(int n)
{
    int cnt = 0;
    for (int i = 1; i <= sqrt(n); i++) {
        if (n % i == 0) {
            // If divisors are equal,
            // count only one
            if (n / i == i)
                cnt++;
 
            else // Otherwise count both
                cnt = cnt + 2;
        }
    }
    return cnt;
}
 
/* Driver program to test above function */
int main()
{
    printf("Total distinct divisors of 100 are : %d",
           countDivisors(100));
    return 0;
}

Java




// JAVA implementation of Naive method
// to count all divisors
import java.io.*;
import java.math.*;
 
class GFG {
 
    // function to count the divisors
    static int countDivisors(int n)
    {
        int cnt = 0;
        for (int i = 1; i <= Math.sqrt(n); i++)
        {
            if (n % i == 0) {
                // If divisors are equal,
                // count only one
                if (n / i == i)
                    cnt++;
 
                else // Otherwise count both
                    cnt = cnt + 2;
            }
        }
        return cnt;
    }
 
    /* Driver program to test above function */
    public static void main(String args[])
    {
        System.out.println("Total distinct "
                  + "divisors of 100 are : " 
                       + countDivisors(100));
    }
}
 
/*This code is contributed by Nikita Tiwari.*/

Python3




# Python3 implementation of Naive method
# to count all divisors
 
import math
 
# function to count the divisors
def countDivisors(n) :
    cnt = 0
    for i in range(1, (int)(math.sqrt(n)) + 1) :
        if (n % i == 0) :
             
            # If divisors are equal,
            # count only one
            if (n / i == i) :
                cnt = cnt + 1
            else : # Otherwise count both
                cnt = cnt + 2
                 
    return cnt
     
# Driver program to test above function */
 
print("Total distinct divisors of 100 are : ",
      countDivisors(100))
 
# This code is contributed by Nikita Tiwari.

C#




// C# implementation of Naive method
// to count all divisors
using System;
 
class GFG {
 
    // function to count the divisors
    static int countDivisors(int n)
    {
        int cnt = 0;
        for (int i = 1; i <= Math.Sqrt(n);
                                      i++)
        {
            if (n % i == 0) {
                 
                // If divisors are equal,
                // count only one
                if (n / i == i)
                    cnt++;
 
                // Otherwise count both
                else
                    cnt = cnt + 2;
            }
        }
         
        return cnt;
    }
 
    // Driver program
    public static void Main()
    {
        Console.WriteLine("Total distinct"
               + " divisors of 100 are : "
                    + countDivisors(100));
    }
}
 
// This code is contributed by anuj_67.

PHP




<?php
// PHP implementation of Naive
// method to count all divisors
 
// function to count the divisors
 
function countDivisors($n)
{
    $cnt = 0;
    for ($i = 1; $i <= sqrt($n); $i++)
    {
        if ($n % $i == 0)
        {
            // If divisors are equal,
            // count only one
            if ($n / $i == $i)
            $cnt++;
 
            // Otherwise count both
            else
                $cnt = $cnt + 2;
        }
    }
    return $cnt;
}
 
// Driver Code
echo "Total distinct divisors of 100 are : ",
        countDivisors(100);
 
// This code is contributed by Ajit
?>

Javascript




<script>
 
// JavaScript program for the above approach
 
    // function to count the divisors
    function countDivisors(n)
    {
        let cnt = 0;
        for (let i = 1; i <= Math.sqrt(n); i++)
        {
            if (n % i == 0) {
                // If divisors are equal,
                // count only one
                if (n / i == i)
                    cnt++;
 
                else // Otherwise count both
                    cnt = cnt + 2;
            }
        }
        return cnt;
    }
 
// Driver Code
     
    document.write("Total distinct "
                  + "divisors of 100 are : " 
                       + countDivisors(100));
             
// This code is contributed by susmitakundugoaldanga.           
</script>

Output : 

Total distinct divisors of 100 are : 9

Optimized Solution (O(n^1/3)) 



For a number N, we try to find a number X ≤ ∛N i.e.  X^3 ≤ N such that it divides the number, and another number Y such that N = X * Y. X consists of all the prime factor of N, which are less than  ∛N and Y contains all the prime factors that are greater. Thus, they have no common factor and their HCF is 1.

We iterate through the numbers 1 to ∛N, and for all primes, we check if the number divides N.  

If the prime is divisible, we divide it as many times as we can from the number N, so that, that specific prime factor no longer remains. We keep doing this for all prime factors less than ∛N. Therefore, the number remaining after the loop won’t have any prime factors less than ∛N.

For N = p1e1 *p2e2*p3e3… where p1, p2, p3.. are the prime factors, the number of divisors is given by (e1+1) * (e2+1) * (e3+1) …

The for loop gives us the product of (e+1) for each prime factor less than ∛N.

The remaining number can only have a maximum of 2 prime factors. We’ll prove this by contradiction.

Assume Y = p1 * p2 * p3 where p1,p2,p3 are prime and p1,p2,p3 > ∛N [Explained above].

Since p1 >∛N and p2 > ∛N and p3 > ∛N



p1*p2*p3 > ∛N*∛N*∛N

=> p1*p2*p3 > N. But Y is a factor of N and cannot be greater than N.

Therefore, there is a contradiction, which implies that one of p1, p2, p3 must be less than ∛N.

But since all primes less than ∛N have been absorbed by X, this is not possible.

So, Y cannot have more than 2 prime factors.

 

Y can therefore have:

1 prime factors if it is prime (Y) with exponent 1

1 prime factors if it is a square of a prime (sqrt(Y)), with exponent 2

2 prime factors if composite (p1, p2) with exponent 1 and 1

Therefore, we multiply:

If Y is prime => (exponent of y .i.e. 1 +1) = 2

If Y is a square of prime => (exponent of sqrt(y) .i.e. 2+1) = 3

If Y is composite => (exponent of p1 +1)*(exponent of p2+1) = 2 * 2 = 4

C++




// C++ program to count distinct divisors
// of a given number n
#include <bits/stdc++.h>
using namespace std;
 
void SieveOfEratosthenes(int n, bool prime[],
                         bool primesquare[], int a[])
{
    // Create a boolean array "prime[0..n]" and
    // initialize all entries it as true. A value
    // in prime[i] will finally be false if i is
    // Not a prime, else true.
    for (int i = 2; i <= n; i++)
        prime[i] = true;
 
    // Create a boolean array "primesquare[0..n*n+1]"
    // and initialize all entries it as false. A value
    // in squareprime[i] will finally be true if i is
    // square of prime, else false.
    for (int i = 0; i <= (n * n + 1); i++)
        primesquare[i] = false;
 
    // 1 is not a prime number
    prime[1] = false;
 
    for (int p = 2; p * p <= n; p++) {
        // If prime[p] is not changed, then
        // it is a prime
        if (prime[p] == true) {
            // Update all multiples of p
            for (int i = p * 2; i <= n; i += p)
                prime[i] = false;
        }
    }
 
    int j = 0;
    for (int p = 2; p <= n; p++) {
        if (prime[p]) {
            // Storing primes in an array
            a[j] = p;
 
            // Update value in primesquare[p*p],
            // if p is prime.
            primesquare[p * p] = true;
            j++;
        }
    }
}
 
// Function to count divisors
int countDivisors(int n)
{
    // If number is 1, then it will have only 1
    // as a factor. So, total factors will be 1.
    if (n == 1)
        return 1;
 
    bool prime[n + 1], primesquare[n * n + 1];
 
    int a[n]; // for storing primes upto n
 
    // Calling SieveOfEratosthenes to store prime
    // factors of n and to store square of prime
    // factors of n
    SieveOfEratosthenes(n, prime, primesquare, a);
 
    // ans will contain total number of distinct
    // divisors
    int ans = 1;
 
    // Loop for counting factors of n
    for (int i = 0;; i++) {
        // a[i] is not less than cube root n
        if (a[i] * a[i] * a[i] > n)
            break;
 
        // Calculating power of a[i] in n.
        int cnt = 1; // cnt is power of prime a[i] in n.
        while (n % a[i] == 0) // if a[i] is a factor of n
        {
            n = n / a[i];
            cnt = cnt + 1; // incrementing power
        }
 
        // Calculating the number of divisors
        // If n = a^p * b^q then total divisors of n
        // are (p+1)*(q+1)
        ans = ans * cnt;
    }
 
    // if a[i] is greater than cube root of n
 
    // First case
    if (prime[n])
        ans = ans * 2;
 
    // Second case
    else if (primesquare[n])
        ans = ans * 3;
 
    // Third case
    else if (n != 1)
        ans = ans * 4;
 
    return ans; // Total divisors
}
 
// Driver Program
int main()
{
    cout << "Total distinct divisors of 100 are : "
         << countDivisors(100) << endl;
    return 0;
}

Java




// JAVA program to count distinct
// divisors of a given number n
import java.io.*;
 
class GFG {
 
    static void SieveOfEratosthenes(int n, boolean prime[],
                                    boolean primesquare[], int a[])
    {
        // Create a boolean array "prime[0..n]" and
        // initialize all entries it as true. A value
        // in prime[i] will finally be false if i is
        // Not a prime, else true.
        for (int i = 2; i <= n; i++)
            prime[i] = true;
 
        /* Create a boolean array "primesquare[0..n*n+1]"
         and initialize all entries it as false.
         A value in squareprime[i] will finally
         be true if i is square of prime,
         else false.*/
        for (int i = 0; i < ((n * n) + 1); i++)
            primesquare[i] = false;
 
        // 1 is not a prime number
        prime[1] = false;
 
        for (int p = 2; p * p <= n; p++) {
            // If prime[p] is not changed,
            // then it is a prime
            if (prime[p] == true) {
                // Update all multiples of p
                for (int i = p * 2; i <= n; i += p)
                    prime[i] = false;
            }
        }
 
        int j = 0;
        for (int p = 2; p <= n; p++) {
            if (prime[p]) {
                // Storing primes in an array
                a[j] = p;
 
                // Update value in
                // primesquare[p*p],
                // if p is prime.
                primesquare[p * p] = true;
                j++;
            }
        }
    }
 
    // Function to count divisors
    static int countDivisors(int n)
    {
        // If number is 1, then it will
        // have only 1 as a factor. So,
        // total factors will be 1.
        if (n == 1)
            return 1;
 
        boolean prime[] = new boolean[n + 1];
        boolean primesquare[] = new boolean[(n * n) + 1];
 
        // for storing primes upto n
        int a[] = new int[n];
 
        // Calling SieveOfEratosthenes to
        // store prime factors of n and to
        // store square of prime factors of n
        SieveOfEratosthenes(n, prime, primesquare, a);
 
        // ans will contain total number
        // of distinct divisors
        int ans = 1;
 
        // Loop for counting factors of n
        for (int i = 0;; i++) {
            // a[i] is not less than cube root n
            if (a[i] * a[i] * a[i] > n)
                break;
 
            // Calculating power of a[i] in n.
            // cnt is power of prime a[i] in n.
            int cnt = 1;
 
            // if a[i] is a factor of n
            while (n % a[i] == 0) {
                n = n / a[i];
 
                // incrementing power
                cnt = cnt + 1;
            }
 
            // Calculating the number of divisors
            // If n = a^p * b^q then total
            // divisors of n are (p+1)*(q+1)
            ans = ans * cnt;
        }
 
        // if a[i] is greater than cube root
        // of n
 
        // First case
        if (prime[n])
            ans = ans * 2;
 
        // Second case
        else if (primesquare[n])
            ans = ans * 3;
 
        // Third case
        else if (n != 1)
            ans = ans * 4;
 
        return ans; // Total divisors
    }
 
    // Driver Program
    public static void main(String args[])
    {
        System.out.println("Total distinct divisors"
                           + " of 100 are : " + countDivisors(100));
    }
}
 
/*This code is contributed by Nikita Tiwari*/

Python3




# Python3 program to count distinct
# divisors of a given number n
 
def SieveOfEratosthenes(n, prime,primesquare, a):
    # Create a boolean array "prime[0..n]"
    # and initialize all entries it as
    # true. A value in prime[i] will finally
    # be false if i is not a prime, else true.
    for i in range(2,n+1):
        prime[i] = True
 
    # Create a boolean array "primesquare[0..n*n+1]"
    # and initialize all entries it as false.
    # A value in squareprime[i] will finally be
    # true if i is square of prime, else false.
    for i in range((n * n + 1)+1):
        primesquare[i] = False
 
    # 1 is not a prime number
    prime[1] = False
 
    p = 2
    while(p * p <= n):
        # If prime[p] is not changed,
        # then it is a prime
        if (prime[p] == True):
            # Update all multiples of p
            i = p * 2
            while(i <= n):
                prime[i] = False
                i += p
        p+=1
     
 
    j = 0
    for p in range(2,n+1):
        if (prime[p]==True):
            # Storing primes in an array
            a[j] = p
 
            # Update value in primesquare[p*p],
            # if p is prime.
            primesquare[p * p] = True
            j+=1
 
# Function to count divisors
def countDivisors(n):
    # If number is 1, then it will
    # have only 1 as a factor. So,
    # total factors will be 1.
    if (n == 1):
        return 1
 
    prime = [False]*(n + 2)
    primesquare = [False]*(n * n + 2)
     
    # for storing primes upto n
    a = [0]*n
 
    # Calling SieveOfEratosthenes to
    # store prime factors of n and to
    # store square of prime factors of n
    SieveOfEratosthenes(n, prime, primesquare, a)
 
    # ans will contain total
    # number of distinct divisors
    ans = 1
 
    # Loop for counting factors of n
    i=0
    while(1):
        # a[i] is not less than cube root n
        if(a[i] * a[i] * a[i] > n):
            break
 
        # Calculating power of a[i] in n.
        cnt = 1 # cnt is power of
                # prime a[i] in n.
        while (n % a[i] == 0): # if a[i] is a factor of n
            n = n / a[i]
            cnt = cnt + 1 # incrementing power
 
        # Calculating number of divisors
        # If n = a^p * b^q then total
        # divisors of n are (p+1)*(q+1)
        ans = ans * cnt
        i+=1
 
    # if a[i] is greater than
    # cube root of n
     
    n=int(n)
    # First case
    if (prime[n]==True):
        ans = ans * 2
 
    # Second case
    elif (primesquare[n]==True):
        ans = ans * 3
 
    # Third case
    elif (n != 1):
        ans = ans * 4
 
    return ans # Total divisors
 
# Driver Code
if __name__=='__main__':
    print("Total distinct divisors of 100 are :",countDivisors(100))
 
# This code is contributed
# by mits

C#




// C# program to count distinct
// divisors of a given number n
using System;
 
class GFG {
 
    static void SieveOfEratosthenes(int n, bool[] prime,
                                    bool[] primesquare, int[] a)
    {
 
        // Create a boolean array "prime[0..n]" and
        // initialize all entries it as true. A value
        // in prime[i] will finally be false if i is
        // Not a prime, else true.
        for (int i = 2; i <= n; i++)
            prime[i] = true;
 
        /* Create a boolean array "primesquare[0..n*n+1]"
        and initialize all entries it as false.
        A value in squareprime[i] will finally
        be true if i is square of prime,
        else false.*/
        for (int i = 0; i < ((n * n) + 1); i++)
            primesquare[i] = false;
 
        // 1 is not a prime number
        prime[1] = false;
 
        for (int p = 2; p * p <= n; p++) {
 
            // If prime[p] is not changed,
            // then it is a prime
            if (prime[p] == true) {
 
                // Update all multiples of p
                for (int i = p * 2; i <= n; i += p)
                    prime[i] = false;
            }
        }
 
        int j = 0;
        for (int p = 2; p <= n; p++) {
            if (prime[p]) {
 
                // Storing primes in an array
                a[j] = p;
 
                // Update value in
                // primesquare[p*p],
                // if p is prime.
                primesquare[p * p] = true;
                j++;
            }
        }
    }
 
    // Function to count divisors
    static int countDivisors(int n)
    {
 
        // If number is 1, then it will
        // have only 1 as a factor. So,
        // total factors will be 1.
        if (n == 1)
            return 1;
 
        bool[] prime = new bool[n + 1];
        bool[] primesquare = new bool[(n * n) + 1];
 
        // for storing primes upto n
        int[] a = new int[n];
 
        // Calling SieveOfEratosthenes to
        // store prime factors of n and to
        // store square of prime factors of n
        SieveOfEratosthenes(n, prime, primesquare, a);
 
        // ans will contain total number
        // of distinct divisors
        int ans = 1;
 
        // Loop for counting factors of n
        for (int i = 0;; i++) {
 
            // a[i] is not less than cube root n
            if (a[i] * a[i] * a[i] > n)
                break;
 
            // Calculating power of a[i] in n.
            // cnt is power of prime a[i] in n.
            int cnt = 1;
 
            // if a[i] is a factor of n
            while (n % a[i] == 0) {
                n = n / a[i];
 
                // incrementing power
                cnt = cnt + 1;
            }
 
            // Calculating the number of divisors
            // If n = a^p * b^q then total
            // divisors of n are (p+1)*(q+1)
            ans = ans * cnt;
        }
 
        // if a[i] is greater than cube root
        // of n
 
        // First case
        if (prime[n])
            ans = ans * 2;
 
        // Second case
        else if (primesquare[n])
            ans = ans * 3;
 
        // Third case
        else if (n != 1)
            ans = ans * 4;
 
        return ans; // Total divisors
    }
 
    // Driver Program
    public static void Main()
    {
        Console.Write("Total distinct divisors"
                      + " of 100 are : " + countDivisors(100));
    }
}
 
// This code is contributed by parashar.

PHP




<?php
// PHP program to count distinct
// divisors of a given number n
 
function SieveOfEratosthenes($n, &$prime,
                             &$primesquare, &$a)
{
    // Create a boolean array "prime[0..n]"
    // and initialize all entries it as
    // true. A value in prime[i] will finally
    // be false if i is not a prime, else true.
    for ($i = 2; $i <= $n; $i++)
        $prime[$i] = true;
 
    // Create a boolean array "primesquare[0..n*n+1]"
    // and initialize all entries it as false.
    // A value in squareprime[i] will finally be
    // true if i is square of prime, else false.
    for ($i = 0; $i <= ($n * $n + 1); $i++)
        $primesquare[$i] = false;
 
    // 1 is not a prime number
    $prime[1] = false;
 
    for ($p = 2; $p * $p <= $n; $p++)
    {
        // If prime[p] is not changed,
        // then it is a prime
        if ($prime[$p] == true)
        {
            // Update all multiples of p
            for ($i = $p * 2;
                 $i <= $n; $i += $p)
                $prime[$i] = false;
        }
    }
 
    $j = 0;
    for ($p = 2; $p <= $n; $p++)
    {
        if ($prime[$p])
        {
            // Storing primes in an array
            $a[$j] = $p;
 
            // Update value in primesquare[p*p],
            // if p is prime.
            $primesquare[$p * $p] = true;
            $j++;
        }
    }
}
 
// Function to count divisors
function countDivisors($n)
{
    // If number is 1, then it will
    // have only 1 as a factor. So,
    // total factors will be 1.
    if ($n == 1)
        return 1;
 
    $prime = array_fill(false, $n + 1, NULL);
    $primesquare = array_fill(false,
                          $n * $n + 1, NULL);
     
    // for storing primes upto n
    $a = array_fill(0, $n, NULL);
  
    // Calling SieveOfEratosthenes to
    // store prime factors of n and to
    // store square of prime factors of n
    SieveOfEratosthenes($n, $prime,
                        $primesquare, $a);
 
    // ans will contain total
    // number of distinct divisors
    $ans = 1;
 
    // Loop for counting factors of n
    for ($i = 0;; $i++)
    {
        // a[i] is not less than cube root n
        if ($a[$i] * $a[$i] * $a[$i] > $n)
            break;
 
        // Calculating power of a[i] in n.
        $cnt = 1; // cnt is power of
                  // prime a[i] in n.
        while ($n % $a[$i] == 0) // if a[i] is a
                                 // factor of n
        {
            $n = $n / $a[$i];
            $cnt = $cnt + 1; // incrementing power
        }
 
        // Calculating the number of divisors
        // If n = a^p * b^q then total
        // divisors of n are (p+1)*(q+1)
        $ans = $ans * $cnt;
    }
 
    // if a[i] is greater than
    // cube root of n
 
    // First case
    if ($prime[$n])
        $ans = $ans * 2;
 
    // Second case
    else if ($primesquare[$n])
        $ans = $ans * 3;
 
    // Third case
    else if ($n != 1)
        $ans = $ans * 4;
 
    return $ans; // Total divisors
}
 
// Driver Code
echo "Total distinct divisors of 100 are : ".
                    countDivisors(100). "\n";
 
// This code is contributed
// by ChitraNayal
?>

Javascript




<script>
 
// Javascript program to count distinct
// divisors of a given number n
     
function SieveOfEratosthenes(n, prime, primesquare, a)
{
     
    // Create a boolean array "prime[0..n]" and
    // initialize all entries it as true. A value
    // in prime[i] will finally be false if i is
    // Not a prime, else true.
    for(let i = 2; i <= n; i++)
        prime[i] = true;
         
    // Create a boolean array "primesquare[0..n*n+1]"
    // and initialize all entries it as false.
    // A value in squareprime[i] will finally
    // be true if i is square of prime,
    // else false.
    for(let i = 0; i < ((n * n) + 1); i++)
        primesquare[i] = false;
 
    // 1 is not a prime number
    prime[1] = false;
 
    for(let p = 2; p * p <= n; p++)
    {
         
        // If prime[p] is not changed,
        // then it is a prime
        if (prime[p] == true)
        {
             
            // Update all multiples of p
            for(let i = p * 2; i <= n; i += p)
                prime[i] = false;
        }
    }
 
    let j = 0;
    for(let p = 2; p <= n; p++)
    {
        if (prime[p])
        {
             
            // Storing primes in an array
            a[j] = p;
 
            // Update value in
            // primesquare[p*p],
            // if p is prime.
            primesquare[p * p] = true;
            j++;
        }
    }
}
 
// Function to count divisors
function countDivisors(n)
{
     
    // If number is 1, then it will
    // have only 1 as a factor. So,
    // total factors will be 1.
    if (n == 1)
        return 1;
 
    let prime = new Array(n + 1);
    let primesquare = new Array((n * n) + 1);
 
    // For storing primes upto n
    let a = new Array(n);
    for(let i = 0; i < n; i++)
    {
        a[i] = 0;
    }
 
    // Calling SieveOfEratosthenes to
    // store prime factors of n and to
    // store square of prime factors of n
    SieveOfEratosthenes(n, prime, primesquare, a);
 
    // ans will contain total number
    // of distinct divisors
    let ans = 1;
 
    // Loop for counting factors of n
    for(let i = 0;; i++)
    {
         
        // a[i] is not less than cube root n
        if (a[i] * a[i] * a[i] > n)
            break;
 
        // Calculating power of a[i] in n.
        // cnt is power of prime a[i] in n.
        let cnt = 1;
 
        // If a[i] is a factor of n
        while (n % a[i] == 0)
        {
            n = n / a[i];
 
            // Incrementing power
            cnt = cnt + 1;
        }
 
        // Calculating the number of divisors
        // If n = a^p * b^q then total
        // divisors of n are (p+1)*(q+1)
        ans = ans * cnt;
    }
 
    // If a[i] is greater than cube root
    // of n
 
    // First case
    if (prime[n])
        ans = ans * 2;
 
    // Second case
    else if (primesquare[n])
        ans = ans * 3;
 
    // Third case
    else if (n != 1)
        ans = ans * 4;
         
    // Total divisors
    return ans;
}
 
// Driver Code
document.write("Total distinct divisors" +
               " of 100 are : " + countDivisors(100));
 
// This code is contributed by avanitrachhadiya2155
 
</script>

Output : 

Total distinct divisors of 100 are : 9

Time Complexity: O(n1/3)

This article is contributed by Karun Anantharaman. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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