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Trick for modular division ( (x1 * x2 …. xn) / b ) mod (m)

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  • Difficulty Level : Hard
  • Last Updated : 05 Aug, 2022
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Given integers x1, x2, x3……xn, b, and m, we are supposed to find the result of ((x1*x2….xn)/b)mod(m). 
Example 1: Suppose that we are required to find (55C5)%(1000000007) i.e ((55*54*53*52*51)/120)%1000000007 
Naive Method : 

  1. Simply calculate the product (55*54*53*52*51)= say x,
  2. Divide x by 120 and then take its modulus with 1000000007

Using Modular Multiplicative Inverse : 
The above method will work only when x1, x2, x3….xn have small values. 
Suppose we are required to find the result where x1, x2, ….xn fall in the range of ~1000000(10^6). So we will have to exploit the rule of modular mathematics which says : 
(a*b)mod(m)=(a(mod(m))*b(mod(m)))mod(m)
Note that the above formula is valid for modular multiplication. A similar formula for division does not exist. 
i.e (a/b)mod(m) != a(mod(m))/b(mod(m)) 

  1. So we are required to find out the modular multiplicative inverse of b say i and then multiply ‘i’ with a .
  2. After this we will have to take the modulus of the result obtained. 
    i.e ((x1*x2….xn)/b)mod(m)=((x1*x2….xn)*i)mod(m)= ((x1)mod(m) * (x2)mod(m) *…. (xn)mod(m) * (i)mod(m))mod(m)

Note: To find modular multiplicative inverse we can use the Extended Euclidean algorithm or Fermat’s Little Theorem. 
Example 2 : Let us suppose that we have to find (55555C5)%(1000000007) i.e ((55555*55554*55553*55552*55551)/120)%1000000007. 
 

C++




#pragma GCC optimize("Ofast")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,avx2,fma")
#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
 
using namespace std;
typedef unsigned long long  ll;
#define int ll
typedef   long double ld;
ll MOD = 998244353;
double eps = 1e-12;
#define mp make_pair
#define pb push_back
#define INF 2e18
#define fast_cin() ios_base::sync_with_stdio(false); cin.tie(NULL); cout.tie(NULL)
 
int modular_inverse(int a, int m){
    for (int x = 1; x < m; x++)
        if (((a%m) * (x%m)) % m == 1)
            return x;
    return 0;
}
 
int32_t main()
{
    // naive method-calculating the result in a single line
     int ans = 1;
     int naive_answer = ((55555 *55554 *55553 *55552 * 55551)/120)% 1000000007;
 
    // modular_inverse() is a user defined function
    // that calculates inverse of a number
     int i = modular_inverse(120, 10000007);
 
    // it will use extended Euclidean algorithm or
    // Fermat’s Little Theorem for calculation.
    // MMI of 120 under division by 1000000007
    // will be 808333339
    for (int i = 0; i < 5; i++)
        ans = (ans * (55555 - i)) % 1000000007;
     
    ans = (ans * i) % 1000000007;
    cout << "Answer using naive method: " << naive_answer << endl;
    cout << "Answer using multiplicative"
         << " modular inverse concept: " << ans;
 
    return 0;
}
 
// The code is contributed by Gautam goel (gautamgoel962)

Java




// Java program to implement the approach
import java.util.*;
 
class GFG {
 
  static int modular_inverse(int a, int m)
  {
    for (int x = 1; x < m; x++)
      if (((a % m) * (x % m)) % m == 1)
        return x;
    return 0;
  }
 
  public static void main(String[] args)
  {
 
    // naive method-calculating the result in a single
    // line
    int ans = 1;
    Long naive_answer = (55555L * 55554L * 55553L
                         * 55552L * 55551L / 120L)
      % 1000000007L;
    System.out.println("Answer using naive method: "
                       + naive_answer);
 
    // modular_inverse() is a user defined function
    // that calculates inverse of a number
    int i = modular_inverse(120, 10000007);
 
    // it will use extended Euclidean algorithm or
    // Fermat’s Little Theorem for calculation.
    // MMI of 120 under division by 1000000007
    // will be 808333339
    for (i = 0; i < 5; i++)
      ans = (ans * (55555 - i)) % 1000000007;
 
    ans = (ans * i) % 1000000007;
 
    System.out.println(
      "Answer using multiplicative modular inverse concept: "
      + ans);
  }
}
 
// The code is contributed by phasing17

Python3




# Python3 program to implement
# the above approach
 
# multiplicative-inverse-under-modulo-m/
def modular_inverse(a, m):
      
    for x in range(1, m):
        if (((a%m) * (x%m)) % m == 1):
            return x
    return -1
 
 
if __name__ == '__main__':
     
    # naive method-calculating the
    # result in a single line
    naive_answer = (((55555 * 55554 *
                      55553 * 55552 *
                      55551) // 120) %
                      1000000007)
 
    ans = 1
 
    # modular_inverse() is a user
    # defined function that calculates
    # inverse of a number
    i = modular_inverse(120, 10000007)
 
    # it will use extended Euclidean
    # algorithm or Fermat's Little
    # Theorem for calculation.
    # MMI of 120 under division by
    # 1000000007 will be 808333339
    for j in range(5):
        ans = ((ans *
               (55555 - j)) %
                1000000007)
 
    ans = (ans * i) % 1000000007
 
    print("Answer using naive method:",
           naive_answer)
    print("Answer using multiplicative" +
          "modular inverse concept:", ans)
 
# This code is contributed by Gautam goel (gautamgoel962)

C#




// C# program to implement the approach
using System;
using System.Numerics;
using System.Collections.Generic;
 
class GFG {
 
    static int modular_inverse(int a, int m)
    {
        for (int x = 1; x < m; x++)
            if (((a % m) * (x % m)) % m == 1)
                return x;
        return 0;
    }
 
    public static void Main(string[] args)
    {
       
        // naive method-calculating the result in a single
        // line
        int ans = 1;
        unchecked
        {
            UInt64 naive_answer
                = ((55555UL * 55554UL * 55553UL * 55552UL
                    * 55551UL)
                   / 120UL)
                  % 1000000007UL;
            Console.WriteLine("Answer using naive method: "
                              + naive_answer);
        }
 
        // modular_inverse() is a user defined function
        // that calculates inverse of a number
        int i = modular_inverse(120, 10000007);
 
        // it will use extended Euclidean algorithm or
        // Fermat’s Little Theorem for calculation.
        // MMI of 120 under division by 1000000007
        // will be 808333339
        for (i = 0; i < 5; i++)
            ans = (ans * (55555 - i)) % 1000000007;
 
        ans = (ans * i) % 1000000007;
 
        Console.WriteLine(
            "Answer using multiplicative modular inverse concept: "
            + ans);
    }
}
 
// The code is contributed by phasing17

Javascript




// JavaScript program to implement above approach
 
// A function to calculate the Modular
// inverse of the function
function modular_inverse(a, m){
    for(let x = 1; x < m; x++){
        if (((a % m) * (x % m)) % m == 1){
                return x;
        }
    }       
}
 
// naive method-calculating the
// result in a single line
let naive_answer =((55555 * 55554 * 55553 * 55552 * 55551) / 120) % 1000000007;
let ans = 1
 
// modular_inverse() is a user
// defined function that calculates
// inverse of a number
let i = modular_inverse(120, 10000007)
 
// it will use extended Euclidean
// algorithm or Fermat's Little
// Theorem for calculation.
// MMI of 120 under division by
// 1000000007 will be 808333339
for(let j = 0;j < 5; j++){
    ans = ((ans * (55555 - j)) % 1000000007)
}
 
ans = (ans * i) % 1000000007
 
console.log("Answer using naive method:",
        naive_answer)
console.log("Answer using multiplicative" +
        "modular inverse concept:", ans)
 
// This code is contributed by Gautam goel (gautamgoel962)

Different Languages give different result for the naive approach. 

C++
Input : 
Output:
    Answer using naive method: 18446744073703577963
    Answer using multiplicative modular inverse concept: 125376140
    
Python 
Input:
Output:
    Answer using naive method: 300820513
    Answer using multiplicative modular inverse concept: 125376140
    
JavaScript:
Input:
Output:
    Answer using naive method: 301201761
    Answer using multiplicative modular inverse concept: 125376140

It is clear from the above example that the naive method will lead to an overflow of data resulting in an incorrect answer. Moreover, using modular inverse will give us the correct answer.
Without Using Modular Multiplicative Inverse : 
But it is interesting to note that a slight change in code will discard the use of finding modular multiplicative inverse. 
 

C++




#include <iostream>
using namespace std;
int main()
{
    long int ans = 1;
    long int mod = (long int)1000000007 * 120;
    for (int i = 0; i < 5; i++)
        ans = (ans * (55555 - i)) % mod;   
    ans = ans / 120;
    cout << "Answer using shortcut: " << ans;
    return 0;
}

Java




class GFG {
 
    public static void main(String[] args)
    {
        long ans = 1;
        long mod = (long)1000000007 * 120;
         
        for (int i = 0; i < 5; i++)
            ans = (ans * (55555 - i)) % mod;
             
        ans = ans / 120;
        System.out.println("Answer using"
                    + " shortcut: "+ ans);
    }
}
 
// This code is contributed by smitha.

Python3




ans = 1
mod = 1000000007 * 120
 
for i in range(0, 5) :
    ans = (ans * (55555 - i)) % mod
     
ans = int(ans / 120)
 
print("Answer using shortcut: ", ans)
 
# This code is contributed by Smitha.

C#




using System;
 
class GFG {
 
    public static void Main()
    {
        long ans = 1;
        long mod = (long)1000000007 * 120;
         
        for (int i = 0; i < 5; i++)
            ans = (ans * (55555 - i)) % mod;
             
        ans = ans / 120;
        Console.Write("Answer using "
                    + "shortcut: "+ ans);
    }
}
 
// This code is contributed by smitha.

PHP




<?php
    $ans = 1;
    $mod = 1000000007 * 120;
     
    for ( $i = 0; $i < 5; $i++)
        $ans = ($ans * (55555 - $i)) %
                                 $mod;
    $ans = $ans / 120;
    echo "Answer using shortcut: " ,
                               $ans;
     
// This code is contributed by anuj_67.
?>

Javascript




<script>
 
var ans = 1;
var mod = 1000000007 * 120;
 
for(var i = 0; i < 5; i++)
    ans = (ans * (55555 - i)) % mod;
     
ans = ans / 120;
document.write("Answer using" +
               " shortcut: "+ ans);
 
// This code is contributed by Kirti
 
</script>

Output

Answer using shortcut: 300820513

Why did it work? 
This will work only in case when the denominator is a factor of numerator i.e. when a % b = 0 following the rule: 
If b | a, then we can write (a/b) % p as (a % p*b)/b. 
This rule proves useful for small values of b.
Let us consider a = x1*x2*x3…….xn 
We have to find ans = (a/b)%1000000007 

  1. Let result of a%(1000000007*b) be y. To avoid overflow, we use modular multiplicative property. This can be represented as 
    a = (1000000007*b)q + y where y < (1000000007*b) and q is an integer
  2. Now dividing LHS and RHS by b, we get 
    y/b = a/b -(1000000007*b)*q/b 
    = a/b -1000000007*q < 1000000007 (From 1) 
    Therefore, y/b is equivalent to (a/b)mod(b*1000000007). 🙂

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