# Find maximum value of x such that n! % (k^x) = 0 in C++

C++Server Side ProgrammingProgramming

Suppose we have two integers n and k. We have to find the maximum value of x, such that n! mod (k^x) = 0. So when n = 5, and k = 2, then output will be 3. As n! = 120, now for different values of x, it will be −

120 mod 2^0 = 0, 120 mod 2^1 = 0, 120 mod 2^2 = 0, 120 mod 2^3 = 0, 120 mod 2^4 = 8, 120 mod 2^5 = 24, 120 mod 2^6 = 56, 120 mod 2^7 = 120. As the max value of x = 3, the result is 0, so the output is 3.

To solve this, we have to follow these steps −

• Take the square root of k, and store it into m
• For i := 2 to m, do the following steps:
• When i = m, then set i := k
• if k is divisible by i, then, divide k by i
• Run a loop to n, and add quotient to a variable called u.
• Store the min value of r, after each loop

## Example

Live Demo

#include <iostream>
#include <cmath>
using namespace std;
int calculateMaxX(int n, int k) {
int result = n, v, u;
int m = sqrt(k) + 1;
for (int i = 2; i <= m && k > 1; i++) {
if (i == m) {
i = k;
}
for (u = v = 0; k % i == 0; v++) {
k /= i;
}
if (v > 0) {
int t = n;
while (t > 0) {
t /= i;
u += t;
}
result = min(result, u / v);
}
}
return result;
}
int main() {
int n = 5;
int k = 2;
cout<<"Maximum value of x is: " << calculateMaxX(n, k);
}

## Output

Maximum value of x is: 3