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Last non-zero digit of a factorial

  • Difficulty Level : Hard
  • Last Updated : 10 Oct, 2021

Given a number n, find the last non-zero digit in n!.
Examples: 
 

Input  : n = 5
Output : 2
5! = 5 * 4 * 3 * 2 * 1 = 120
Last non-zero digit in 120 is 2.

Input  : n = 33
Output : 4

 

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A Simple Solution is to first find n!, then find the last non-zero digit of n. This solution doesn’t work for even slightly large numbers due to arithmetic overflow.
A Better Solution is based on the below recursive formula 

Let D(n) be the last non-zero digit in n!
If tens digit (or second last digit) of n is odd
    D(n) = 4 * D(floor(n/5)) * D(Unit digit of n) 
If tens digit (or second last digit) of n is even
    D(n) = 6 * D(floor(n/5)) * D(Unit digit of n)

Illustration of the formula: 
For the numbers less than 10 we can easily find the last non-zero digit by the above simple solution, i.e., first computing n!, then finding the last digit. 
D(1) = 1, D(2) = 2, D(3) = 6, D(4) = 4, D(5) = 2, 
D(6) = 2, D(7) = 4, D(8) = 2, D(9) = 8.
 



D(1) to D(9) are assumed to be precomputed.

Example 1: n = 27 [Second last digit is even]:
D(27) = 6 * D(floor(27/5)) * D(7)
      = 6 * D(5) * D(7)
      = 6 * 2 * 4 
      = 48
Last non-zero digit is  8

Example 2: n = 33 [Second last digit is odd]:
D(33) = 4 * D(floor(33/5)) * D(3)
      = 4 * D(6) * 6
      = 4 * 2 * 6
      = 48
Last non-zero digit is  8

How does the above formula work? 
The below explanation provides intuition behind the formula. Readers may refer  Refer http://math.stackexchange.com/questions/130352/last-non-zero-digit-of-a-factorial for complete proof.
 

14! = 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 
                     6 * 5 * 4 * 3 * 2 * 1

Since we are asked about last non-zero digit, 
we remove all 5's and equal number of 2's from
factors of 14!.  We get following:

14! = 14 * 13 * 12 * 11 * 2 * 9 * 8 * 7 *
                           6 * 3 * 2 * 1

Now we can get last non-zero digit by multiplying
last digits of above factors!

In n! a number of 2’s are always more than a number of 5’s. To remove trailing 0’s, we remove 5’s and equal number of 2’s. 
Let a = floor(n/5), b = n % 5. After removing an equal number of 5’s and 2’s, we can reduce the problem from n! to 2a * a! * b! 
D(n) = 2a * D(a) * D(b)
Implementation: 
 

C++




// C++ program to find last non-zero digit in n!
#include<bits/stdc++.h>
using namespace std;
 
// Initialize values of last non-zero digit of
// numbers from 0 to 9
int dig[] = {1, 1, 2, 6, 4, 2, 2, 4, 2, 8};
 
int lastNon0Digit(int n)
{
     if (n < 10)
        return dig[n];
 
    // Check whether tens (or second last) digit
    // is odd or even
    // If n = 375, So n/10 = 37 and (n/10)%10 = 7
    // Applying formula for even and odd cases.
    if (((n/10)%10)%2 == 0)
        return (6*lastNon0Digit(n/5)*dig[n%10]) % 10;
    else
        return (4*lastNon0Digit(n/5)*dig[n%10]) % 10;
}
 
// Driver code
int main()
{
    int n = 14;
    cout << lastNon0Digit(n);
    return 0;
}

Java




// Java program to find last
// non-zero digit in n!
 
class GFG
{
    // Initialize values of last non-zero digit of
    // numbers from 0 to 9
    static int dig[] = {1, 1, 2, 6, 4, 2, 2, 4, 2, 8};
     
    static int lastNon0Digit(int n)
    {
        if (n < 10)
            return dig[n];
     
        // Check whether tens (or second last)
        // digit is odd or even
        // If n = 375, So n/10 = 37 and
        // (n/10)%10 = 7 Applying formula for
        // even and odd cases.
        if (((n / 10) % 10) % 2 == 0)
            return (6 * lastNon0Digit(n / 5)
                    * dig[n % 10]) % 10;
        else
            return (4 * lastNon0Digit(n / 5)
                    * dig[n % 10]) % 10;
    }
     
    // Driver code
    public static void main (String[] args)
    {
        int n = 14;
        System.out.print(lastNon0Digit(n));
    }
}
// This code is contributed by Anant Agarwal.

Python3




# Python program to find
# last non-zero digit in n!
 
# Initialize values of
# last non-zero digit of
# numbers from 0 to 9
dig= [1, 1, 2, 6, 4, 2, 2, 4, 2, 8]
  
def lastNon0Digit(n):
    if (n < 10):
        return dig[n]
  
     # Check whether tens (or second last) digit
     # is odd or even
     # If n = 375, So n/10 = 37 and (n/10)%10 = 7
     # Applying formula for even and odd cases.
    if (((n//10)%10)%2 == 0):
        return (6*lastNon0Digit(n//5)*dig[n%10]) % 10
    else:
        return (4*lastNon0Digit(n//5)*dig[n%10]) % 10
    return 0
 
# driver code
n = 14
 
print(lastNon0Digit(n))
 
# This code is contributed
# by Anant Agarwal.

C#




// C# program to find last
// non-zero digit in n!
using System;
 
class GFG {
     
    // Initialize values of last non-zero
    // digit of numbers from 0 to 9
    static int []dig = {1, 1, 2, 6, 4, 2, 2, 4, 2, 8};
     
    static int lastNon0Digit(int n)
    {
        if (n < 10)
            return dig[n];
     
        // Check whether tens (or second
        // last) digit is odd or even
        // If n = 375, So n/10 = 37 and
        // (n/10)%10 = 7 Applying formula
        // for even and odd cases.
        if (((n / 10) % 10) % 2 == 0)
            return (6 * lastNon0Digit(n / 5) *
                    dig[n % 10]) % 10;
        else
            return (4 * lastNon0Digit(n / 5) *
                    dig[n % 10]) % 10;
    }
     
    // Driver code
    public static void Main ()
    {
        int n = 14;
        Console.Write(lastNon0Digit(n));
    }
}
 
// This code is contributed by Nitin Mittal.

PHP




<?php
// PHP program to find last
// non-zero digit in n!
 
// Initialize values of
// last non-zero digit of
// numbers from 0 to 9
$dig = array(1, 1, 2, 6, 4,
             2, 2, 4, 2, 8);
 
function lastNon0Digit($n)
{
     
    global $dig;
    if ($n < 10)
        return $dig[$n];
 
    // Check whether tens(or second 
    // last) digit is odd or even
    // If n = 375, So n/10 = 37 and
    // (n/10)%10 = 7
    // Applying formula for even
    // and odd cases.
    if ((($n / 10) % 10) % 2 == 0)
        return (6 * lastNon0Digit($n / 5) *
                       $dig[$n % 10]) % 10;
    else
        return (4 * lastNon0Digit($n / 5) *
                        $dig[$n % 10]) % 10;
}
 
// Driver code
$n = 14;
echo(lastNon0Digit($n));
 
// This code is contributed by Ajit.
?>

Javascript




<script>
 
    // Javascript program to find
    // last non-zero digit in n!
     
    // Initialize values of last non-zero
    // digit of numbers from 0 to 9
    let dig = [1, 1, 2, 6, 4, 2, 2, 4, 2, 8];
       
    function lastNon0Digit(n)
    {
        if (n < 10)
            return dig[n];
       
        // Check whether tens (or second
        // last) digit is odd or even
        // If n = 375, So n/10 = 37 and
        // (n/10)%10 = 7 Applying formula
        // for even and odd cases.
        if ((parseInt(n / 10, 10) % 10) % 2 == 0)
         return (6 * lastNon0Digit(parseInt(n / 5, 10))
            * dig[n % 10]) % 10;
        else
         return (4 * lastNon0Digit(parseInt(n / 5, 10))
            * dig[n % 10]) % 10;
    }
     
    let n = 14;
      document.write(lastNon0Digit(n));
     
</script>
Output
2

 A Simple Solution based on recursion having worst-case Time Complexity O(nLog(n)).

Approach:-

  1. It is given that you have to find the last positive digit. Now a digit is made multiple of 10 if there are 2 and 5. They produce a number with last digit 0.
  2. Now what we can do is divide each array element into its shortest divisible form by 5 and increase count of such occurrences.
  3. Now divide each array element into its shortest divisible form by 2 and decrease count of such occurrences. This way we are not considering the multiplication of 2 and a 5 in our multiplication(number of 2’s present in multiplication result upto n is always more than number 0f 5’s).
  4. Multiply each number(after removing pairs of 2’s and 5’s) now and store just last digit by taking remainder by 10.
  5. Now call recursively for smaller numbers by (currentNumber – 1) as parameter.

Below is the implementation of the above approach: 

Java




/*package whatever //do not write package name here */
 
import java.io.*;
 
class GFG {
    // Helper Function to return the rightmost non-zero
    // digit
    public static void
    callMeFactorialLastDigit(int n, int[] result,
                             int sumOf5)
    {
        int number = n; // assaigning to new variable.
        if (number == 1)
            return; // base case
 
        // To store the count of times 5 can
        // divide the number.
        while (number % 5 == 0) {
            number /= 5;
            // increase count of 5
            sumOf5++;
        }
 
        // Divide the number by
        // 2 as much as possible
        while (sumOf5 != 0 && (number & 1) == 0) {
            number >>= 1; // dividing the number by 2
            sumOf5--;
        }
 
        /*multiplied result and current number(after
        removing pairs) and do modular division to get the
        last digit of the resultant number.*/
        result[0] = (result[0] * (number % 10)) % 10;
        // calling again for (currentNumber - 1)
        callMeFactorialLastDigit(n - 1, result, sumOf5);
    }
 
    public static int lastNon0Digit(int n)
    {
        int[] result = { 1 }; // single element array.
        callMeFactorialLastDigit(n, result, 0);
        return result[0];
    }
 
    public static void main(String[] args)
    {
        System.out.println(lastNon0Digit(7)); // 3040
        System.out.println(lastNon0Digit(12)); // 479001600
    }
}
//This code is contributed by KaaL-EL.
Output
4
6

we used single element array (int[] result = {1}) instead of integer as Java is Strictly Pass by Value!. It does not allow pass by reference for primitive data types. That’s why I used a single element array so that the recursive function can change the value of variable(result here). If we would have taken (int result = 1) then this variable remain unaffected.

This article is contributed by Niteesh kumar & KaaL-EL. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 




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