Related Articles
Find maximum power of a number that divides a factorial

Given two numbers, fact and n, find the largest power of n that divides fact! (Factorial of fact).

Examples:

```Input :
fact = 5, n = 2
Output :
3
Explanation:
Value of 5! is 120. The largest power
of 2 that divides 120 is 8 (or 23

Input :
fact = 146, n = 15
Output :
35```

The idea is based on Legendre’s formula which finds largest power of a prime number that divides fact!. We find all prime factors of n. For every prime factor we find largest power of it that divides fact!. Finally we return minimum of all found powers.

Illustration :

```fact = 146, n=15
First find the prime factor of 15 that are 3
and 5 then first divide with 3 and add i.e.

Applying Legendre’s formula for prime
factor 3.
[146/3]+[48/3]+[16/3]+[5/3]+[1/3] = 70
48  +   16  +  5  +  1  +  0   = 70
There is 70 is maximum power of 3 prime number.
146! is divisible by 3^70 which is maximum.

Applying Legendre’s formula for prime
factor 5.
[146/5]+[29/5]+[5/5]+[1/5] = 35
29  +   5  +  1  +  0   = 35
There is 35 is maximum power of 5 prime
number.```

Minimum of two powers is 35 which is our answer.
Note : If multiple powers of a prime factor are present in n, then we divide the count to get the maximum power value for this factor.

Below is the implementation of the above approach:

## C++

 `// CPP program to find largest power of``// a number (which may be composite) that``// divides factorial.``#include ``using` `namespace` `std;` `// for find maximum power of prime number``// p which can divide fact number``int` `findPowerPrime(``int` `fact, ``int` `p)``{``    ``int` `res = 0;``    ``while` `(fact > 0) {``        ``res += fact / p;``        ``fact /= p;``    ``}` `    ``return` `res;``}` `// Returns sum of all factors of n.``int` `findPowerComposite(``int` `fact, ``int` `n)``{``    ``// To store result (minimum power of a``    ``// prime factor that divides fact! )``    ``int` `res = INT_MAX;` `    ``// Traverse through all prime factors``    ``// of n.``    ``for` `(``int` `i = 2; i <= ``sqrt``(n); i++) {` `        ``// counter for count the``        ``// power of prime number``        ``int` `count = 0;``        ``while` `(n % i == 0) {``            ``count++;``            ``n = n / i;``        ``}` `        ``if` `(count > 0) {` `            ``// Maximum power of i that divides``            ``// fact!. We divide by count to handle``            ``// multiple occurrences of a prime factor.``            ``int` `curr_pow = findPowerPrime(fact, i) / count;``            ``res = min(res, curr_pow);``        ``}``    ``}` `    ``// This condition is to handle``    ``// the case when n is a prime``    ``// number greater than 2.``    ``if` `(n >= 2) {``        ``int` `curr_pow = findPowerPrime(fact, n);``        ``res = min(res, curr_pow);``    ``}` `    ``return` `res;``}` `// Driver code``int` `main()``{``    ``int` `fact = 146, n = 5;``    ` `    ``// Function Call``    ``cout << findPowerComposite(fact, n);``    ``return` `0;``}`

## Java

 `// Java program to find largest power of``// a number (which may be composite) that``// divides factorial.``class` `GFG {` `    ``// for find maximum power of prime number``    ``// p which can divide fact number``    ``static` `int` `findPowerPrime(``int` `fact, ``int` `p)``    ``{``        ``int` `res = ``0``;``        ``while` `(fact > ``0``) {``            ``res += fact / p;``            ``fact /= p;``        ``}` `        ``return` `res;``    ``}` `    ``// Returns sum of all factors of n.``    ``static` `int` `findPowerComposite(``int` `fact, ``int` `n)``    ``{``      ` `        ``// To store result (minimum power of a``        ``// prime factor that divides fact! )``        ``int` `res = Integer.MAX_VALUE;` `        ``// Traverse through all prime factors``        ``// of n.``        ``for` `(``int` `i = ``2``; i <= Math.sqrt(n); i++)``        ``{` `            ``// counter for count the``            ``// power of prime number``            ``int` `count = ``0``;``            ``if` `(n % i == ``0``)``            ``{``                ``count++;``                ``n = n / i;``            ``}` `            ``if` `(count > ``0``)``            ``{` `                ``// Maximum power of i that divides``                ``// fact!. We divide by count to``                ``// handle multiple occurrences of``                ``// a prime factor.``                ``int` `curr_pow``                    ``= findPowerPrime(fact, i) / count;``                ``res = Math.min(res, curr_pow);``            ``}``        ``}` `        ``// This condition is to handle``        ``// the case when n is a prime``        ``// number greater than 2.``        ``if` `(n >= ``2``) {``            ``int` `curr_pow = findPowerPrime(fact, n);``            ``res = Math.min(res, curr_pow);``        ``}` `        ``return` `res;``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `fact = ``146``, n = ``5``;``      ` `        ``// Function Call``        ``System.out.println(findPowerComposite(fact, n));``    ``}``}` `// This code is contributed by prerna saini`

## Python3

 `# Python program to find largest power of``# a number (which may be composite) that``# divides factorial.``import` `math` `# For find maximum power of prime number``# p which can divide fact number`  `def` `findPowerPrime(fact, p):``    ``res ``=` `0``    ``while` `fact:``        ``res ``+``=` `fact ``/``/` `p``        ``fact ``/``/``=` `p` `    ``return` `res` `# Returns sum of all factors of n`  `def` `findPowerComposite(fact, n):` `    ``# To store result (minimum power of a``    ``# prime factor that divides fact! )``    ``res ``=` `math.inf` `    ``# Traverse through all prime factors``    ``# of n.``    ``for` `i ``in` `range``(``2``, ``int``(n``*``*``0.5``) ``+` `1``):` `        ``# Counter for count the``        ``# power of prime number``        ``count ``=` `0``        ``if` `not` `n ``%` `i:``            ``count ``+``=` `1``            ``n ``=` `n ``/``/` `i` `        ``if` `count:` `            ``# Maximum power of i that divides``            ``# fact!. We divide by count to handle``            ``# multiple occurrences of a prime factor.``            ``curr_pow ``=` `findPowerPrime(fact, i) ``/``/` `count``            ``res ``=` `min``(res, curr_pow)` `    ``# This condition is to handle``    ``# the case when n is a prime``    ``# number greater than 2.``    ``if` `n >``=` `2``:``        ``curr_pow ``=` `findPowerPrime(fact, n)``        ``res ``=` `min``(res, curr_pow)` `    ``return` `res`  `# Driver code``fact ``=` `146``n ``=` `5` `# Function Call``print``(findPowerComposite(fact, n))`  `# This code is contributed by Ansu Kumari`

## C#

 `// C# program to find largest power of``// a number (which may be composite) that``// divides factorial.``using` `System;` `class` `GFG {` `    ``// for find maximum power of prime number``    ``// p which can divide fact number``    ``static` `int` `findPowerPrime(``int` `fact, ``int` `p)``    ``{``        ``int` `res = 0;``        ``while` `(fact > 0) {``            ``res += fact / p;``            ``fact /= p;``        ``}` `        ``return` `res;``    ``}` `    ``// Returns sum of all factors of n.``    ``static` `int` `findPowerComposite(``int` `fact, ``int` `n)``    ``{``        ``// To store result (minimum power of a``        ``// prime factor that divides fact! )``        ``int` `res = ``int``.MaxValue;` `        ``// Traverse through all prime factors``        ``// of n.``        ``for` `(``int` `i = 2; i <= Math.Sqrt(n); i++) {` `            ``// counter for count the``            ``// power of prime number``            ``int` `count = 0;``            ``if` `(n % i == 0) {``                ``count++;``                ``n = n / i;``            ``}` `            ``if` `(count > 0) {` `                ``// Maximum power of i that divides``                ``// fact!. We divide by count to``                ``// handle multiple occurrences of``                ``// a prime factor.``                ``int` `curr_pow``                    ``= findPowerPrime(fact, i) / count;``                ``res = Math.Min(res, curr_pow);``            ``}``        ``}` `        ``// This condition is to handle``        ``// the case when n is a prime``        ``// number greater than 2.``        ``if` `(n >= 2) {``            ``int` `curr_pow = findPowerPrime(fact, n);``            ``res = Math.Min(res, curr_pow);``        ``}` `        ``return` `res;``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{``        ``int` `fact = 146, n = 5;``      ` `        ``// Function Call``        ``Console.WriteLine(findPowerComposite(fact, n));``    ``}``}``// This code is contributed by vt_m`

## PHP

 ` 0)``    ``{       ``        ``\$res` `+= ``intval``(``\$fact` `/ ``\$p``);``        ``\$fact` `= ``intval``(``\$fact` `/ ``\$p``);``    ``}``    ``return` `\$res``;``}``  ` `// Returns sum of``// all factors of n.``function` `findPowerComposite(``\$fact``, ``\$n``)``{``    ``// To store result (minimum``    ``// power of a prime factor``    ``// that divides fact! )``    ``\$res` `= PHP_INT_MAX;``  ` `    ``// Traverse through all``    ``// prime factors of n.``    ``for` `(``\$i` `= 2;``         ``\$i` `<= sqrt(``\$n``); ``\$i``++)``    ``{       ``  ` `        ``// counter for count the``        ``// power of prime number``        ``\$count` `= 0;``        ``if` `(``\$n` `% ``\$i` `== 0)``        ``{``            ``\$count``++;``            ``\$n` `= ``intval``(``\$n` `/ ``\$i``);``        ``}``  ` `        ``if` `(``\$count` `> 0)``        ``{``  ` `            ``// Maximum power of i``            ``// that divides fact!.``            ``// We divide by count``            ``// to handle multiple``            ``// occurrences of a``            ``// prime factor.``            ``\$curr_pow` `= ``intval``(findPowerPrime(``                               ``\$fact``, ``\$i``) / ``\$count``);``            ``\$res` `= min(``\$res``, ``\$curr_pow``);``         ``}``    ``}``  ` `    ``// This condition is to``    ``// handle the case when``    ``// n is a prime number``    ``// greater than 2.``    ``if` `(``\$n` `>= 2)``    ``{``        ``\$curr_pow`  `= findPowerPrime(``\$fact``, ``\$n``);``        ``\$res` `= min(``\$res``, ``\$curr_pow``);``    ``}``    ``return` `\$res``;``}``  ` `// Driver code``\$fact` `= 146; ``\$n` `= 5;` `// Function Call``echo` `(findPowerComposite(``\$fact``, ``\$n``));` `// This code is contributed by``// Manish Shaw(manishshaw1)``?>`

## Javascript

 ``
Output
`35`

Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

My Personal Notes arrow_drop_up