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Find the last digit when factorial of A divides factorial of B

We are given two numbers A and B such that B >= A. We need to compute the last digit of this resulting F such that F = B!/A! where 1 = A, B <= 10^18 (A and B are very large).

Examples:

Input : A = 2, B = 4
Output : 2
Explanation : A! = 2 and B! = 24. 
F = 24/2 = 12 --> last digit = 2

Input : 107 109
Output : 2

As we know, factorial function grows on an exponential rate. Even the largest data type
cannot hold factorial of numbers like 100. To compute factorial of moderately large numbers, refer this.
Here the given constraints are very large. Thus, calculating the two factorials and later
dividing them and computing the last digit is practically an impossible task.

Thus we have to find an alternate approach to break down our problem. It is known that the last digit of factorial always belongs to the set {0, 1, 2, 4, 6}
The approach is as follows: –
1) We evaluate the difference between B and A
2) If the (B – A) >= 5, then the answer is always 0
3) If the difference (B – A) < 5, then we iterate from (A+1) to B, multiply and store them. multiplication_answer % 10 shall be our answer.

C++

// CPP program to find last digit of a number 
// obtained by dividing factorial of a number
// with factorial of another number.
#include <iostream>
using namespace std;
  
// Function which computes the last digit
// of resultant of B!/A!
int computeLastDigit(long long int A, long long int B)
{
  
    int variable = 1;
    if (A == B) // If A = B, B! = A! and B!/A! = 1
        return 1;
  
    // If difference (B - A) >= 5, answer = 0
    else if ((B - A) >= 5) 
        return 0;
  
    else {
  
        // If non of the conditions are true, we
        // iterate from  A+1 to B and multiply them. 
        // We are only concerned for the last digit,
        // thus we take modulus of 10
        for (long long int i = A + 1; i <= B; i++)
            variable = (variable * (i % 10)) % 10;
  
        return variable % 10;
    }
}
  
// driver function
int main()
{
    cout << computeLastDigit(2632, 2634);
    return 0;
}

Java

// Java program to find last digit of a number 
// obtained by dividing factorial of a number
// with factorial of another number.
import java.io.*;
  
class GFG {
  
// Function which computes the last digit
// of resultant of B!/A!
static int computeLastDigit(long A, long B)
{
  
    int variable = 1;
    if (A == B) // If A = B, B! = A! and B!/A! = 1
        return 1;
  
    // If difference (B - A) >= 5, answer = 0
    else if ((B - A) >= 5
        return 0;
  
    else {
  
        // If non of the conditions are true, we
        // iterate from A+1 to B and multiply them. 
        // We are only concerned for the last digit,
        // thus we take modulus of 10
        for (long i = A + 1; i <= B; i++)
            variable = (int)(variable * (i % 10)) % 10;
  
        return variable % 10;
    }
}
  
// driver function
public static void main(String[] args)
{
    System.out.println(computeLastDigit(2632, 2634));
}
}
  
// This article is contributed by Prerna Saini

Python3

# Python program to find
# last digit of a number 
# obtained by dividing
# factorial of a number
# with factorial of another number.
  
# Function which computes
# the last digit
# of resultant of B!/A!
def computeLastDigit(A,B):
  
    variable = 1
    if (A == B): # If A = B, B! = A! and B!/A! = 1
        return 1
   
    # If difference (B - A) >= 5, answer = 0
    elif ((B - A) >= 5): 
        return 0
   
    else
   
        # If non of the conditions
        # are true, we
        # iterate from  A+1 to B
        # and multiply them. 
        # We are only concerned
        # for the last digit,
        # thus we take modulus of 10
        for i in range(A + 1, B + 1):
            variable = (variable * (i % 10)) % 10
   
        return variable % 10
      
# driver function
  
print(computeLastDigit(2632, 2634))
  
# This code is contributed
# by Anant Agarwal.

C#

// C# program to find last digit of
// a number obtained by dividing 
// factorial of a number with
// factorial of another number.
using System;
  
class GFG {
  
// Function which computes the last 
// digit of resultant of B!/A!
static int computeLastDigit(long A, long B)
{
  
    int variable = 1;
     // If A = B, B! = A! 
     // and B!/A! = 1
     if (A == B)
        return 1;
  
    // If difference (B - A) >= 5,
    // answer = 0
    else if ((B - A) >= 5) 
        return 0;
  
    else {
  
        // If non of the conditions are true, we
        // iterate from A+1 to B and multiply them. 
        // We are only concerned for the last digit,
        // thus we take modulus of 10
        for (long i = A + 1; i <= B; i++)
            variable = (int)(variable * 
                       (i % 10)) % 10;
  
        return variable % 10;
    }
}
  
// Driver Code
public static void Main()
{
    Console.WriteLine(computeLastDigit(2632, 2634));
}
}
  
// This code is contributed by vt_m.

PHP

<?php
// PHP program to find last digit of a number 
// obtained by dividing factorial of a number
// with factorial of another number.
  
// Function which computes the last 
// digit of resultant of B!/A!
function computeLastDigit($A, $B)
{
  
    $variable = 1;
      
    // If A = B, B! = A! 
    // and B!/A! = 1
    if ($A == $B
        return 1;
  
    // If difference (B - A) >= 5,
    // answer = 0
    else if (($B - $A) >= 5) 
        return 0;
  
    else
    {
  
        // If non of the conditions 
        // are true, we iterate from
        // A+1 to B and multiply them. 
        // We are only concerned for
        // the last digit, thus we 
        // take modulus of 10
        for ($i = $A + 1; $i <= $B; $i++)
            $variable = ($variable * ($i % 10)) % 10;
  
        return $variable % 10;
    }
}
  
    // Driver Code
    echo computeLastDigit(2632, 2634);
  
// This code is contributed by ajit
?>


Output:

2

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.



This article is attributed to GeeksforGeeks.org

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