Finding powers of any number P in N!

Prerequisite: Print all prime factors and their powers
Given natural numbers N and P, the task is to find the power of P in the factorization of N!.

Examples

Input: N = 4, P = 2
Output: 3
Explanation:
Power of 2 in the prime factorization of 4! = 24 is 3

Input: N = 24, P = 4
Output: 11

Naive Approach: The idea is to find the power of P for each number from 1 to N and add them as we know during multiplication power is added.
Time Complexity: O(N*P)



Efficient Approach:
To find the power of the number P in N! do the following:

  1. Find all the Prime Factors of the number P with their frequency by using the approach discussed in this article. Store the Prime Factors with their frequency in map.
  2. Find the power of every Prime Factors of P in the factorization of N! by using the approach discussed in this article.
  3. Divide the every power obtained in the above steps by their corresponding frequency in the map.
  4. Store the result of above steps in an array and minimum of those element will give the power of P in the factorisation of N!.

Below is the implementation of the above approach:

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ program to find the power
// of P in N!
#include <bits/stdc++.h>
using namespace std;
  
// Map to store all the prime
// factors of P
unordered_map<int, int> Map;
  
// Function to find the prime
// factors of N im Map
void findPrimeFactors(int N)
{
    int i;
  
    // Clear map
    Map.clear();
  
    // Check for factors of 2
    while (N % 2 == 0) {
        Map[2] += 1;
        N /= 2;
    }
  
    // Find all the prime factors
    for (i = 3; i <= sqrt(N); i += 2) {
  
        // If i is a factors
        // then increase the
        // frequency of i
        while (N % i == 0) {
            Map[i] += 1;
            N /= i;
        }
    }
  
    if (N > 2) {
        Map[N] += 1;
    }
}
  
// Function to find the power
// of prime number P in N!
int PowInFactN(int N, int P)
{
    int ans = 0;
    int temp = P;
  
    // Loop until temp <= N
    while (temp <= N) {
  
        // Add the number of
        // numbers divisible
        // by N
        ans += N / temp;
  
        // Each time multiply
        // temp by P
        temp = temp * P;
    }
  
    // Returns ans
    return ans;
}
  
// Function that find the
// powers of any P in N!
int findPowers(int N, int P)
{
  
    // Find all prime factors
    // of number P
    findPrimeFactors(P);
  
    // To store the powers of
    // all prime factors
    vector<int> Powers;
  
    // Traverse the map
    for (auto& it : Map) {
  
        // Prime factor and
        // corres. powers
        int primeFac = it.first;
        int facPow = it.second;
  
        // Find power of prime
        // factor primeFac
        int p = PowInFactN(N,
                           primeFac);
  
        // Divide frequency by
        // facPow
        p /= facPow;
  
        // Store the power of
        // primeFac^facPow
        Powers.push_back(p);
    }
  
    // Return the minimum
    // element in Power array
    return *min_element(Powers.begin(),
                        Powers.end());
}
  
// Driver's Code
int main()
{
    int N = 24, P = 4;
  
    // Function to find power of
    // P in N!
    cout << findPowers(N, P);
    return 0;
}

chevron_right


Java

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java program to find the power
// of P in N!
import java.util.*;
  
class GFG{
   
// Map to store all the prime
// factors of P
static HashMap<Integer,Integer> Map = new HashMap<Integer,Integer>();
   
// Function to find the prime
// factors of N im Map
static void findPrimeFactors(int N)
{
    int i;
   
    // Clear map
    Map.clear();
   
    // Check for factors of 2
    while (N % 2 == 0) {
        if(Map.containsKey(2))
            Map.put(2, Map.get(2) + 1);
        else
            Map.put(2, 1);
        N /= 2;
    }
   
    // Find all the prime factors
    for (i = 3; i <= Math.sqrt(N); i += 2) {
   
        // If i is a factors
        // then increase the
        // frequency of i
        while (N % i == 0) {
            if(Map.containsKey(i))
                Map.put(i, Map.get(i) + 1);
            else
                Map.put(i, 1);
            N /= i;
        }
    }
   
    if (N > 2) {
        if(Map.containsKey(N))
            Map.put(N, Map.get(N) + 1);
        else
            Map.put(N, 1);
    }
}
   
// Function to find the power
// of prime number P in N!
static int PowInFactN(int N, int P)
{
    int ans = 0;
    int temp = P;
   
    // Loop until temp <= N
    while (temp <= N) {
   
        // Add the number of
        // numbers divisible
        // by N
        ans += N / temp;
   
        // Each time multiply
        // temp by P
        temp = temp * P;
    }
   
    // Returns ans
    return ans;
}
   
// Function that find the
// powers of any P in N!
static int findPowers(int N, int P)
{
   
    // Find all prime factors
    // of number P
    findPrimeFactors(P);
   
    // To store the powers of
    // all prime factors
    Vector<Integer> Powers = new Vector<Integer>();
   
    // Traverse the map
    for (Map.Entry<Integer, Integer> it : Map.entrySet()) {
   
        // Prime factor and
        // corres. powers
        int primeFac = it.getKey();
        int facPow = it.getValue();
   
        // Find power of prime
        // factor primeFac
        int p = PowInFactN(N,
                           primeFac);
   
        // Divide frequency by
        // facPow
        p /= facPow;
   
        // Store the power of
        // primeFac^facPow
        Powers.add(p);
    }
   
    // Return the minimum
    // element in Power array
    return Collections.min(Powers);
}
   
// Driver's Code
public static void main(String[] args)
{
    int N = 24, P = 4;
   
    // Function to find power of
    // P in N!
    System.out.print(findPowers(N, P));
}
}
  
// This code is contributed by Rajput-Ji

chevron_right


C#

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# program to find the power
// of P in N!
using System;
using System.Linq;
using System.Collections.Generic;
  
class GFG{
  
// Map to store all the prime
// factors of P
static Dictionary<int,int> Map = new Dictionary<int,int>();
  
// Function to find the prime
// factors of N im Map
static void findPrimeFactors(int N)
{
    int i;
  
    // Clear map
    Map.Clear();
  
    // Check for factors of 2
    while (N % 2 == 0) {
        if(Map.ContainsKey(2))
            Map[2] = Map[2] + 1;
        else
            Map.Add(2, 1);
        N /= 2;
    }
  
    // Find all the prime factors
    for (i = 3; i <= Math.Sqrt(N); i += 2) {
  
        // If i is a factors
        // then increase the
        // frequency of i
        while (N % i == 0) {
            if(Map.ContainsKey(i))
                Map[i] = Map[i] + 1;
            else
                Map.Add(i, 1);
            N /= i;
        }
    }
  
    if (N > 2) {
        if(Map.ContainsKey(N))
            Map[N] =Map[N] + 1;
        else
            Map.Add(N, 1);
    }
}
  
// Function to find the power
// of prime number P in N!
static int PowInFactN(int N, int P)
{
    int ans = 0;
    int temp = P;
  
    // Loop until temp <= N
    while (temp <= N) {
  
        // Add the number of
        // numbers divisible
        // by N
        ans += N / temp;
  
        // Each time multiply
        // temp by P
        temp = temp * P;
    }
  
    // Returns ans
    return ans;
}
  
// Function that find the
// powers of any P in N!
static int findPowers(int N, int P)
{
  
    // Find all prime factors
    // of number P
    findPrimeFactors(P);
  
    // To store the powers of
    // all prime factors
    List<int> Powers = new List<int>();
  
    // Traverse the map
    foreach (KeyValuePair<int, int> it in Map) {
  
        // Prime factor and
        // corres. powers
        int primeFac = it.Key;
        int facPow = it.Value;
  
        // Find power of prime
        // factor primeFac
        int p = PowInFactN(N,
                        primeFac);
  
        // Divide frequency by
        // facPow
        p /= facPow;
  
        // Store the power of
        // primeFac^facPow
        Powers.Add(p);
    }
  
    // Return the minimum
    // element in Power array
    return Powers.Min();
}
  
// Driver's Code
public static void Main(String[] args)
{
    int N = 24, P = 4;
  
    // Function to find power of
    // P in N!
    Console.Write(findPowers(N, P));
}
}
  
// This code is contributed by sapnasingh4991

chevron_right


Output:

11

Time Complexity: O(sqrt(P)*(logP N))

Don’t stop now and take your learning to the next level. Learn all the important concepts of Data Structures and Algorithms with the help of the most trusted course: DSA Self Paced. Become industry ready at a student-friendly price.




My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.