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A problem on factorials

4 replies [Last post]
sanjeev's picture
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Joined: 21 Feb 2011

I started solving easy problems in Perl from codechef.com

This is my first solution which got accepted. So, I am sharing with you guys. You can post your solution in Perl or in any programming languages.

 

Problem

The most important part of a GSM network is so called Base Transceiver Station (BTS). These transceivers form the areas called cells (this term gave the name to the cellular phone) and every phone connects to the BTS with the strongest signal (in a little simplified view). Of course, BTSes need some attention and technicians need to check their function periodically.

The technicians faced a very interesting problem recently. Given a set of BTSes to visit, they needed to find the shortest path to visit all of the given points and return back to the central company building. Programmers have spent several months studying this problem but with no results. They were unable to find the solution fast enough. After a long time, one of the programmers found this problem in a conference article. Unfortunately, he found that the problem is so called "Traveling Salesman Problem" and it is very hard to solve. If we have N BTSes to be visited, we can visit them in any order, giving us N! possibilities to examine. The function expressing that number is called factorial and can be computed as a product 1.2.3.4....N. The number is very high even for a relatively small N.

The programmers understood they had no chance to solve the problem. But because they have already received the research grant from the government, they needed to continue with their studies and produce at least some results. So they started to study behavior of the factorial function.

For example, they defined the function Z. For any positive integer N, Z(N) is the number of zeros at the end of the decimal form of number N!. They noticed that this function never decreases. If we have two numbers N1<N2, then Z(N1) <= Z(N2). It is because we can never "lose" any trailing zero by multiplying by any positive number. We can only get new and new zeros. The function Z is very interesting, so we need a computer program that can determine its value efficiently.

 

Input

There is a single positive integer T on the first line of input (equal to about 100000). It stands for the number of numbers to follow. Then there are T lines, each containing exactly one positive integer number N, 1 <= N <= 1000000000.

 

Output

For every number N, output a single line containing the single non-negative integer Z(N).

Example

Sample Input:

6
3
60
100
1024
23456
8735373

Sample Output:

0
14
24
253
5861
2183837




and time limit is 8 seconds. My solution took 2.92 seconds

 

Solution

  1. use strict;
  2. use warnings;
  3. use integer;
  4.  
  5. my $x;
  6. chomp($x = <STDIN>);
  7.  
  8. while($x--){
  9. my $y;
  10. print "\n";
  11. chomp($y = <STDIN>);
  12. my $count =1;
  13. my $sum =0;
  14.  
  15. while($y/5**$count >= 1 )
  16. {
  17. $sum += $y/5**$count;
  18. $count++;
  19. }
  20. print $sum;
  21. }

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Joined: 22 Feb 2011
Re: A problem on factorials

Congrts... for first accepted solution..

You can make that a bit fast by removing  "while($y/5**$count >= 1 ) &  $sum += $y/5**$count " by single variable. Actually you are doing same calculation twice.

One more thing if we are using CPP and program takes large number of input and output than it is better to go with printf & scanf instead of cin & cout.

 

sanjeev's picture
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Joined: 21 Feb 2011
Re: its saying time limit exceeded

Hi Abhishek,
I observed that after submitting the code. But you know I changed the code and submitted again.

This time time limit got exceeded.

Does its output depend upon the internet connection also?

I just changed few lines

previous lines

while($y/5**$count){
sum+=$y/5**$count;
 

modified lines
$factor = 5**$count;
        while($y/$factor >= 1 )
        {
            $sum += $y/$factor;

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Offline
Joined: 22 Feb 2011
Re: its saying time limit exceeded

No as per my knowledge execution time does not depend on the internet connection.

Here this my CPP  code which got accepted in 0.32sec.

Code

#include <cstdio>
#include <iostream>
 
using namespace std;
 
int main(){
        int T;
        scanf("%d",&T);
        while(T--){
                long long int mul5=5;
                int n;
                scanf("%d",&n);
                long long int ans=0;
                while(mul5<=n){
                         ans+=n/mul5;
                         mul5*=5;
                }
                printf("%lld\n",ans);
        }
        return 0;
}    

 

 

sanjeev's picture
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Joined: 21 Feb 2011
It seems C and C++ codes work

It seems C and C++ codes work much faster than Perl code  on their compiler. Although its not the case in general.

You can test it by benchmarking both the codes and run at the same environment at your workstation.

Well Abhishek, your code looks much neater and effective than mine.
Thanks for sharing.

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