# Smallest factorial number binary search

QUESTION

Given a number n. The task is to find the smallest number whose factorial contains at least n trailing zeroes.\n\nExamples:\n\nInput : n = 1\nOutput : 5 \n1!, 2!, 3!, 4! does not contain trailing zero.\n5! = 120, which contains one trailing zero.\n\nInput : n = 6\nOutput : 25\n25! = 15511210043330985984000000\n\nInput:\nThe first line of input contains an integer T denoting the no of test cases. Then T test cases follow. Each test case contains an integer n.\n\nOutput:\nFor each test case in a new line print the required output.\n\nConstraints:\n1<=T<=100\n1<=n<=100\n.

``````// C++ program tofind smallest number whose
// factorial contains at least n trailing
// zeroes.
#include<bits/stdc++.h>
using namespace std;

// Return true if number's factorial contains
// at least n trailing zero else false.
bool check(int p, int n)
{
int temp = p, count = 0, f = 5;
while (f <= temp)
{
count += temp/f;
f = f*5;
}
return (count >= n);
}

// Return smallest number whose factorial
// contains at least n trailing zeroes
int findNum(int n)
{
// If n equal to 1, return 5.
// since 5! = 120.
if (n==1)
return 5;

// Initalising low and high for binary
// search.
int low = 0;
int high = 5*n;

// Binary Search.
while (low <high)
{
int mid = (low + high) >> 1;

// Checking if mid's factorial contains
// n trailing zeroes.
if (check(mid, n))
high = mid;
else
low = mid+1;
}

return low;
}

// driver code
int main()
{
int t, n, i ;
cin>>t;

for(i=1;i<=t;i++)
{   cin>>n;
cout << findNum(n) << " ";
}
return 0;

}
``````