Given an integer n and a prime number p, find the largest x such that px (p raised to power x) divides n! (factorial)
Examples:
Input: n = 7, p = 3 Output: x = 2 32 divides 7! and 2 is the largest such power of 3. Input: n = 10, p = 3 Output: x = 4 34 divides 10! and 4 is the largest such power of 3.
n! is multiplication of {1, 2, 3, 4, …n}.
How many numbers in {1, 2, 3, 4, ….. n} are divisible by p?
Every p’th number is divisible by p in {1, 2, 3, 4, ….. n}. Therefore in n!, there are ?n/p? numbers divisible by p. So we know that the value of x (largest power of p that divides n!) is at-least ?n/p?.
Can x be larger than ?n/p? ?
Yes, there may be numbers which are divisible by p2, p3, …
How many numbers in {1, 2, 3, 4, ….. n} are divisible by p2, p3, …?
There are ?n/(p2)? numbers divisible by p2 (Every p2‘th number would be divisible). Similarly, there are ?n/(p3)? numbers divisible by p3 and so on.
What is the largest possible value of x?
So the largest possible power is ?n/p? + ?n/(p2)? + ?n/(p3)? + ……
Note that we add only ?n/(p2)? only once (not twice) as one p is already considered by expression ?n/p?. Similarly, we consider ?n/(p3)? (not thrice).
Below is implementation of above idea.
C++
// C++ program to find largest x such that p*x divides n! #include <iostream> using namespace std; // Returns largest power of p that divides n! int largestPower(int n, int p) { // Initialize result int x = 0; // Calculate x = n/p + n/(p^2) + n/(p^3) + .... while (n) { n /= p; x += n; } return x; } // Driver code int main() { int n = 10, p = 3; cout << "The largest power of "<< p << " that divides " << n << "! is "<< largestPower(n, p) << endl; return 0; } // This code is contributed by shubhamsingh10 |
C
// C program to find largest x such that p*x divides n! #include <stdio.h> // Returns largest power of p that divides n! int largestPower(int n, int p) { // Initialize result int x = 0; // Calculate x = n/p + n/(p^2) + n/(p^3) + .... while (n) { n /= p; x += n; } return x; } // Driver program int main() { int n = 10, p = 3; printf("The largest power of %d that divides %d! is %d\n", p, n, largestPower(n, p)); return 0; } |
Java
// Java program to find largest x such that p*x divides n! import java.io.*; class GFG { // Function that returns largest power of p // that divides n! static int Largestpower(int n, int p) { // Initialize result int ans = 0; // Calculate x = n/p + n/(p^2) + n/(p^3) + .... while (n > 0) { n /= p; ans += n; } return ans; } // Driver program public static void main (String[] args) { int n = 10; int p = 3; System.out.println(" The largest power of " + p + " that divides " + n + "! is " + Largestpower(n, p)); } } |
Python3
# Python3 program to find largest # x such that p*x divides n! # Returns largest power of p that divides n! def largestPower(n, p): # Initialize result x = 0 # Calculate x = n/p + n/(p^2) + n/(p^3) + .... while n: n /= p x += n return x # Driver program n = 10; p = 3print ("The largest power of %d that divides %d! is %d\n"% (p, n, largestPower(n, p))) # This code is contributed by Shreyanshi Arun. |
C#
// C# program to find largest x // such that p * x divides n! using System; public class GFG { // Function that returns largest // power of p that divides n! static int Largestpower(int n, int p) { // Initialize result int ans = 0; // Calculate x = n / p + n / (p ^ 2) + // n / (p ^ 3) + .... while (n > 0) { n /= p; ans += n; } return ans; } // Driver Code public static void Main () { int n = 10; int p = 3; Console.Write(" The largest power of " + p + " that divides " + n + "! is " + Largestpower(n, p)); } } // This code is contributed by Sam007 |
PHP
<?php // PHP program to find largest // x such that p*x divides n! // Returns largest power // of p that divides n! function largestPower($n, $p) { // Initialize result $x = 0; // Calculate x = n/p + // n/(p^2) + n/(p^3) + .... while ($n) { $n = (int)$n / $p; $x += $n; } return floor($x); } // Driver Code $n = 10; $p = 3; echo "The largest power of ", $p ; echo " that divides ",$n , "! is "; echo largestPower($n, $p); // This code is contributed by ajit ?> |
Output:
The largest power of 3 that divides 10! is 4
Time complexity of the above solution is Logpn.
What to do if p is not prime?
We can find all prime factors of p and compute result for every prime factor. Refer Largest power of k in n! (factorial) where k may not be prime for details.
Source:
http://e-maxx.ru/algo/factorial_divisors
This article is contributed by Ankur. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
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