First digit in factorial of a number

Last Updated : 27 Jan, 2023

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)

// 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.

First digit in factorial of a number

ajay0007

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First digit in factorial of a number

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