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Find prime factors of Z such that Z is product of all even numbers till N that are product of two distinct prime numbers

  • Last Updated : 02 Aug, 2021

Given a number N (N > 6), the task is to print the prime factorization of a number Z, where Z is the product of all numbers ≤ N that are even and can be expressed as the product of two distinct prime numbers. 

Example:

Input: N = 6
Output: 2→1
               3→1
Explanation: 6 is the only number ≤ N, which is even and a product of two distinct prime numbers (2 and 3). Therefore, Z=6. 
Now, prime factorization of Z=2×3

Input: N = 5
Output: 2→2
               3→1
               5→1
Explanation: The only even numbers ≤N, which can be expressed as the product of two distinct prime numbers, are 6 (2×3) and 10 (2×5). Therefore, Z = 6*10=60 = 2x2x3x5

 

Observation: The following observation helps to solve the problem:



  1. Since the required numbers need to be even and product of two distinct prime numbers, they will be of the form 2×P, where P is a prime number ≤ N / 2.
  2. Thus, the prime factorization of Z will be of the form 2x.31.51…P1, where P is the last prime number ≤ N/2 and X is the number of prime numbers in the range [3, N / 2].

Approach: Follow the steps to solve the problem:

  1. Store all prime numbers ≤ N / 2, using Sieve of Eratosthenes in a vector, say prime.
  2. Store the number of primes in the range [3,  N/2] in a variable, say x.
  3. Print the prime factorization, where the coefficient of 2 is x and the coefficients of all other primes in the range [3, N/2] is 1.

Below is the implementation of the above approach:

C++




// C++ implementation for the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to print the prime factorization of the product
// of all numbers <=N that are even and can be expressed as a
// product of two distinct prime numbers
void primeFactorization(int N)
{
    // sieve of Eratosthenese
    int sieve[N / 2 + 1] = { 0 };
    for (int i = 2; i <= N / 2; i++) {
        if (sieve[i] == 0) {
            for (int j = i * i; j <= N / 2; j += i) {
                sieve[j] = 1;
            }
        }
    }
 
    // Store prime numbers in the range [3, N/2]
    vector<int> prime;
    for (int i = 3; i <= N / 2; i++)
        if (sieve[i] == 0)
            prime.push_back(i);
 
    // print the coefficient of 2 in the prime
    // factorization
    int x = prime.size();
    cout << "2->" << x << endl;
 
    // print the coefficients of other primes
    for (int i : prime)
        cout << i << "->1" << endl;
}
// Driver code
int main()
{
    // Input
    int N = 18;
 
    // Function calling
    primeFactorization(N);
    return 0;
}

Java




// Java implementation of
// the above approach
import java.util.*;
import java.util.HashMap;
 
class GFG{
             
// Function to print the prime factorization
// of the product of all numbers <=N that are
// even and can be expressed as a product of
// two distinct prime numbers
static void primeFactorization(int N)
{
     
    // Sieve of Eratosthenese
    int[] sieve = new int[N / 2 + 1];
    for(int i = 0; i <= N / 2; i++)
    {
        sieve[i] = 0;
    }
    for(int i = 2; i <= N / 2; i++)
    {
        if (sieve[i] == 0)
        {
            for(int j = i * i; j <= N / 2; j += i)
            {
                sieve[j] = 1;
            }
        }
    }
   
    // Store prime numbers in the range [3, N/2]
    ArrayList<Integer> prime = new ArrayList<Integer>();
    for(int i = 3; i <= N / 2; i++)
        if (sieve[i] == 0)
            prime.add(i);
   
    // Print the coefficient of 2 in the prime
    // factorization
    int x = prime.size();
    System.out.println("2->" + x);
   
    // Print the coefficients of other primes
    for(int i : prime)
        System.out.println(i + "->1");
}
 
// Driver Code
public static void main(String args[])
{
     
    // Input
    int N = 18;
   
    // Function calling
    primeFactorization(N);
}
}
 
// This code is contributed by sanjoy_62

Python3




# Python3 implementation for the above approach
 
# Function to print the prime factorization
# of the product of all numbers <=N that are
# even and can be expressed as a product of
# two distinct prime numbers
def primeFactorization(N):
     
    # Sieve of Eratosthenese
    sieve = [0 for i in range(N // 2 + 1)]
    for i in range(2, N // 2 + 1, 1):
        if (sieve[i] == 0):
            for j in range(i * i, N // 2 + 1, i):
                sieve[j] = 1
 
    # Store prime numbers in the range [3, N/2]
    prime = []
    for i in range(3, N // 2 + 1, 1):
        if (sieve[i] == 0):
            prime.append(i)
 
    # Print the coefficient of 2 in the
    # prime factorization
    x = len(prime)
    print("2->", x)
 
    # Print the coefficients of other primes
    for i in prime:
        print(i, "->1")
 
# Driver code
if __name__ == '__main__':
     
    # Input
    N = 18
 
    # Function calling
    primeFactorization(N)
     
# This code is contributed by ipg2016107

C#




// C# implementation of
// the above approach
using System;
using System.Collections.Generic;
 
class GFG{
 
// Function to print the prime factorization
// of the product of all numbers <=N that are
// even and can be expressed as a product of
// two distinct prime numbers
static void primeFactorization(int N)
{
     
    // sieve of Eratosthenese
    int[] sieve = new int[N / 2 + 1];
    for(int i = 0; i <= N / 2; i++)
    {
        sieve[i] = 0;
    }
    for(int i = 2; i <= N / 2; i++)
    {
        if (sieve[i] == 0)
        {
            for (int j = i * i; j <= N / 2; j += i)
            {
                sieve[j] = 1;
            }
        }
    }
  
    // Store prime numbers in the range [3, N/2]
    List<int> prime = new List<int>();
    for(int i = 3; i <= N / 2; i++)
        if (sieve[i] == 0)
            prime.Add(i);
  
    // Print the coefficient of 2 in the prime
    // factorization
    int x = prime.Count;
    Console.WriteLine("2->" + x);
  
    // Print the coefficients of other primes
    foreach(int i in prime)
        Console.WriteLine(i + "->1");
}
 
// Driver Code
public static void Main(String[] args)
{
     
    // Input
    int N = 18;
  
    // Function calling
    primeFactorization(N);
}
}
 
// This code is contributed by avijitmondal1998

Javascript




<script>
 
// JavaScript program for the above approach
 
// Function to print the prime factorization
// of the product of all numbers <=N that are
// even and can be expressed as a product of
// two distinct prime numbers
function primeFactorization(N)
{
     
    // Sieve of Eratosthenese
    let sieve = new Array(parseInt(N / 2) + 1).fill(0);
    for(let i = 2; i <= N / 2; i++)
    {
        if (sieve[i] == 0)
        {
            for(let j = i * i; j <= N / 2; j += i)
            {
                sieve[j] = 1;
            }
        }
    }
 
    // Store prime numbers in the range [3, N/2]
    let prime = [];
    for(let i = 3; i <= N / 2; i++)
        if (sieve[i] == 0)
            prime.push(i);
 
    // Print the coefficient of 2 in the prime
    // factorization
    let x = prime.length;
    document.write("2->" + x);
    document.write("<br>")
 
    // Print the coefficients of other primes
    for(let i of prime)
    {
        document.write(i + "->1");
        document.write("<br>");
    }
}
 
// Driver code
 
// Input
let N = 18;
 
// Function calling
primeFactorization(N);
 
// This code is contributed by Potta Lokesh
 
</script>
Output
2->3
3->1
5->1
7->1

Time Complexity: O(N * log(log(N)))
Auxiliary Space: O(N)

 

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