# Find maximum power of a number that divides a factorial in C++

Suppose we have two numbers n and fact. We have to find the largest power of n, that divides fact! (factorial of fact). So if fact = 5, and n = 2, then output will be 3. So 5! = 120, and this is divisible by 2^3 = 8.

Here we will use the Legendre’s formula. This finds largest power of a prime, that divides fact!. We will find all prime factors of n, then find largest power of it, that divides fact!.

So if fact is 146, and n = 15, then prime factors of n are 5 and 3. So

for 3, it will be [146/3] + [48/3] + [16/3] + [5/3] + [1/3] = 48 + 16 + 5 + 1 + 0 = 70.

for 5, it will be [146/5] + [29/5] + [5/5] + [1/3] = 29 + 5 + 1 + 0 = 35.

## Example

#include<iostream>
#include<cmath>
using namespace std;
int getPowerPrime(int fact, int p) {
int res = 0;
while (fact > 0) {
res += fact / p;
fact /= p;
}
return res;
}
int findMinPower(int fact, int n) {
int res = INT_MAX;
for (int i = 2; i <= sqrt(n); i++) {
int cnt = 0;
if (n % i == 0) {
cnt++;
n = n / i;
}
if (cnt > 0) {
int curr = getPowerPrime(fact, i) / cnt;
res = min(res, curr);
}
}
if (n >= 2) {
int curr = getPowerPrime(fact, n);
res = min(res, curr);
}
return res;
}
int main() {
int fact = 146, n = 5;
cout << "Minimum power: " << findMinPower(fact, n);
}

## Output

Minimum power: 35