First digit in factorial of a number

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Given a positive integer n, find the first digit in its factorial.

Examples :

Input  : n = 5
Output : 1
Factorial of 5 is 120 and first
digit is 1.

Input  : 1000
Output : 4

A simple solution is to compute factorial of number, then find first digit in it.
The above solution causes overflow soon. A better solution is to use the fact that factorial contains trailing 0s and removing trailing 0s does not change first digit. For example, first digit of x * y is same as x * y * 100 for x > 0 and y > 0.

C++

// A C++ program for finding the First digit
// of the large factorial number
#include <bits/stdc++.h>
using namespace std;
 
int firstDigit(int n)
{
    long long int fact = 1;
 
    for (int i = 2; i <= n; i++) {
        fact = fact * i;
 
        // Removing trailing 0s as this
        // does not change first digit.
        while (fact % 10 == 0) 
            fact = fact / 10;        
    }
 
    // loop for divide the fact until it 
    // become the single digit and return
    // the fact
    while (fact >= 10) 
        fact = fact / 10;
     
    return fact;
}
 
// derive main
int main()
{
    int n = 5;
    cout << firstDigit(n);
    return 0;
}

Java

// A Java program for finding the First digit
// of the large factorial number
class GFG{
static int firstDigit(int n)
{
    int fact = 1;
 
    for (int i = 2; i <= n; i++) {
        fact = fact * i;
 
        // Removing trailing 0s as this
        // does not change first digit.
        while (fact % 10 == 0) 
            fact = fact / 10;     
    }
 
    // loop for divide the fact until it 
    // become the single digit and return
    // the fact
    while (fact >= 10) 
        fact = fact / 10;
     
    return fact;
}
 
// derive main
public static void main(String[] args)
{
    int n = 5;
    System.out.println(firstDigit(n));
}
}
//This code is contributed by Smitha Dinesh Semwal

Python3

# Python3 program for finding 
# the First digit of the 
# large factorial number
import math
def firstDigit(n) :
    fact = 1
 
    for i in range(2, n + 1) :
        fact = fact * i
 
        # Removing trailing 0s 
        # as this does not 
        # change first digit.
        while (fact % 10 == 0) : 
            fact = int(fact / 10)
 
    # loop for divide the fact
    # until it become the single
    # digit and return the fact
 
    while (fact >= 10) :
        fact = int(fact / 10)
     
    return math.floor(fact)
 
# Driver Code
n = 5
print (firstDigit(n))
 
# This code is contributed by 
# Manish Shaw(manishshaw1)

C#

// A C# program for finding the First digit
// of the large factorial number
using System;
 
class GFG {
     
    static int firstDigit(int n)
    {
        int fact = 1;
     
        for (int i = 2; i <= n; i++)
        {
            fact = fact * i;
     
            // Removing trailing 0s as this
            // does not change first digit.
            while (fact % 10 == 0) 
                fact = fact / 10;     
        }
     
        // loop for divide the fact until
        // it become the single digit and
        // return the fact
        while (fact >= 10) 
            fact = fact / 10;
         
        return fact;
    }
     
    // driver function
    public static void Main()
    {
        int n = 5;
         
        Console.Write(firstDigit(n));
    }
}
 
// This code is contributed by parashar.

PHP

<?php
// PHP program for finding 
// the First digit of the 
// large factorial number
 
function firstDigit($n)
{
    $fact = 1;
 
    for ($i = 2; $i <= $n; $i++) 
    {
        $fact = $fact * $i;
 
        // Removing trailing 0s as this
        // does not change first digit.
        while ($fact % 10 == 0) 
            $fact = $fact / 10; 
    }
 
    // loop for divide the fact
    // until it become the single
    // digit and return    the fact
 
    while ($fact >= 10) 
        $fact = $fact / 10;
     
    return floor($fact);
}
 
// Driver Code
$n = 5;
echo firstDigit($n);
 
// This code is contributed by aj_36.
?>

Javascript

<script>
 
// JavaScript program for finding the
// First digit of the large factorial number
function firstDigit(n)
{
    let fact = 1;
   
    for(let i = 2; i <= n; i++) 
    {
        fact = fact * i;
   
        // Removing trailing 0s as this
        // does not change first digit.
        while (fact % 10 == 0) 
            fact = fact / 10;     
    }
   
    // Loop for divide the fact until it 
    // become the single digit and return
    // the fact
    while (fact >= 10) 
        fact = fact / 10;
       
    return (Math.round(fact));
}
 
// Driver code
let n = 5;
 
document.write(firstDigit(n));
 
// This code is contributed by splevel62
 
</script>

Output :

Time Complexity: O(n * log10(n)), where log10(n) time is to remove trailing zeroes and it is running n times
Auxiliary Space: O(1)

The above code also fails for slightly higher values. The best idea seems to be to find factorial of large number and then find first digit.

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