Chinese Remainder Theorem | Set 2 (Inverse Modulo based Implementation)

We are given two arrays num[0..k-1] and rem[0..k-1]. In num[0..k-1], every pair is coprime (gcd for every pair is 1). We need to find minimum positive number x such that:


     x % num[0]    =  rem[0], 
     x % num[1]    =  rem[1], 
     .......................
     x % num[k-1]  =  rem[k-1] 

Example:

Input:  num[] = {3, 4, 5}, rem[] = {2, 3, 1}
Output: 11
Explanation: 
11 is the smallest number such that:
  (1) When we divide it by 3, we get remainder 2. 
  (2) When we divide it by 4, we get remainder 3.
  (3) When we divide it by 5, we get remainder 1. 

We strongly recommend to refer below post as a prerequisite for this.

Chinese Remainder Theorem | Set 1 (Introduction)

We have discussed a Naive solution to find minimum x. In this article, an efficient solution to find x is discussed.



The solution is based on below formula.


x =  ( ∑ (rem[i]*pp[i]*inv[i]) ) % prod
   Where 0 <= i <= n-1

rem[i] is given array of remainders

prod is product of all given numbers
prod = num[0] * num[1] * ... * num[k-1]

pp[i] is product of all divided by num[i]
pp[i] = prod / num[i]

inv[i] = Modular Multiplicative Inverse of 
         pp[i] with respect to num[i]

Example:

Let us take below example to understand the solution
   num[] = {3, 4, 5}, rem[] = {2, 3, 1}
   prod  = 60 
   pp[]  = {20, 15, 12}
   inv[] = {2,  3,  3}  // (20*2)%3 = 1, (15*3)%4 = 1
                        // (12*3)%5 = 1

   x = (rem[0]*pp[0]*inv[0] + rem[1]*pp[1]*inv[1] + 
        rem[2]*pp[2]*inv[2]) % prod
     = (2*20*2 + 3*15*3 + 1*12*3) % 60
     = (40 + 135 + 36) % 60
     = 11

Refer this for nice visual explanation of above formula.

Below is the implementation of above formula. We can use Extended Euclid based method discussed here to find inverse modulo.

C++

// A C++ program to demonstrate working of Chinise remainder
// Theorem
#include<bits/stdc++.h>
using namespace std;

// Returns modulo inverse of a with respect to m using extended
// Euclid Algorithm. Refer below post for details:
// https://www.geeksforgeeks.org/multiplicative-inverse-under-modulo-m/
int inv(int a, int m)
{
    int m0 = m, t, q;
    int x0 = 0, x1 = 1;

    if (m == 1)
       return 0;

    // Apply extended Euclid Algorithm
    while (a > 1)
    {
        // q is quotient
        q = a / m;

        t = m;

        // m is remainder now, process same as
        // euclid's algo
        m = a % m, a = t;

        t = x0;

        x0 = x1 - q * x0;

        x1 = t;
    }

    // Make x1 positive
    if (x1 < 0)
       x1 += m0;

    return x1;
}

// k is size of num[] and rem[].  Returns the smallest
// number x such that:
//  x % num[0] = rem[0],
//  x % num[1] = rem[1],
//  ..................
//  x % num[k-2] = rem[k-1]
// Assumption: Numbers in num[] are pairwise coprime
// (gcd for every pair is 1)
int findMinX(int num[], int rem[], int k)
{
    // Compute product of all numbers
    int prod = 1;
    for (int i = 0; i < k; i++)
        prod *= num[i];

    // Initialize result
    int result = 0;

    // Apply above formula
    for (int i = 0; i < k; i++)
    {
        int pp = prod / num[i];
        result += rem[i] * inv(pp, num[i]) * pp;
    }

    return result % prod;
}

// Driver method
int main(void)
{
    int num[] = {3, 4, 5};
    int rem[] = {2, 3, 1};
    int k = sizeof(num)/sizeof(num[0]);
    cout << "x is " << findMinX(num, rem, k);
    return 0;
}

Java

// A Java program to demonstrate 
// working of Chinise remainder 
// Theorem
import java.io.*;

class GFG {
    
    // Returns modulo inverse of a 
    // with respect to m using extended
    // Euclid Algorithm. Refer below post for details:
    // https://www.geeksforgeeks.org/multiplicative-inverse-under-modulo-m/
    static int inv(int a, int m)
    {
        int m0 = m, t, q;
        int x0 = 0, x1 = 1;
    
        if (m == 1)
        return 0;
    
        // Apply extended Euclid Algorithm
        while (a > 1)
        {
            // q is quotient
            q = a / m;
    
            t = m;
    
            // m is remainder now, process
            // same as euclid's algo
            m = a % m;a = t;
    
            t = x0;
    
            x0 = x1 - q * x0;
    
            x1 = t;
        }
    
        // Make x1 positive
        if (x1 < 0)
        x1 += m0;
    
        return x1;
    }
    
    // k is size of num[] and rem[].
    // Returns the smallest number
    // x such that:
    // x % num[0] = rem[0],
    // x % num[1] = rem[1],
    // ..................
    // x % num[k-2] = rem[k-1]
    // Assumption: Numbers in num[] are pairwise 
    // coprime (gcd for every pair is 1)
    static int findMinX(int num[], int rem[], int k)
    {
        // Compute product of all numbers
        int prod = 1;
        for (int i = 0; i < k; i++)
            prod *= num[i];
    
        // Initialize result
        int result = 0;
    
        // Apply above formula
        for (int i = 0; i < k; i++)
        {
            int pp = prod / num[i];
            result += rem[i] * inv(pp, num[i]) * pp;
        }
    
        return result % prod;
    }
    
    // Driver method
    public static void main(String args[])
    {
        int num[] = {3, 4, 5};
        int rem[] = {2, 3, 1};
        int k = num.length;
        System.out.println("x is " +findMinX(num, rem, k));
    }
}

// This code is contributed by nikita Tiwari.

Python3

# A Python 3program to demonstrate
# working of Chinise remainder
# Theorem

# Returns modulo inverse of a with
# respect to m using extended
# Euclid Algorithm. Refer below 
# post for details:
# https://www.geeksforgeeks.org/multiplicative-inverse-under-modulo-m/
def inv(a, m) :
    
    m0 = m
    x0 = 0
    x1 = 1

    if (m == 1) :
        return 0

    # Apply extended Euclid Algorithm
    while (a > 1) :
        # q is quotient
        q = a // m

        t = m

        # m is remainder now, process 
        # same as euclid's algo
        m = a % m
        a = t

        t = x0

        x0 = x1 - q * x0

        x1 = t
    
    # Make x1 positive
    if (x1 < 0) :
        x1 = x1 + m0

    return x1

# k is size of num[] and rem[]. 
# Returns the smallest
# number x such that:
# x % num[0] = rem[0],
# x % num[1] = rem[1],
# ..................
# x % num[k-2] = rem[k-1]
# Assumption: Numbers in num[] 
# are pairwise coprime
# (gcd for every pair is 1)
def findMinX(num, rem, k) :
    
    # Compute product of all numbers
    prod = 1
    for i in range(0, k) :
        prod = prod * num[i]

    # Initialize result
    result = 0

    # Apply above formula
    for i in range(0,k):
        pp = prod // num[i]
        result = result + rem[i] * inv(pp, num[i]) * pp
    
    
    return result % prod

# Driver method
num = [3, 4, 5]
rem = [2, 3, 1]
k = len(num)
print( "x is " , findMinX(num, rem, k))

# This code is contributed by Nikita Tiwari.

C#


// A C# program to demonstrate 
// working of Chinese remainder 
// Theorem
using System;

class GFG
{
    // Returns modulo inverse of 
    // 'a' with respect to 'm' 
    // using extended Euclid Algorithm. 
    // Refer below post for details:
    // https://www.geeksforgeeks.org/multiplicative-inverse-under-modulo-m/
    static int inv(int a, int m)
    {
        int m0 = m, t, q;
        int x0 = 0, x1 = 1;
    
        if (m == 1)
        return 0;
    
        // Apply extended 
        // Euclid Algorithm
        while (a > 1)
        {
            // q is quotient
            q = a / m;
    
            t = m;
    
            // m is remainder now, 
            // process same as 
            // euclid's algo
            m = a % m; a = t;
    
            t = x0;
    
            x0 = x1 - q * x0;
    
            x1 = t;
        }
    
        // Make x1 positive
        if (x1 < 0)
        x1 += m0;
    
        return x1;
    }
    
    // k is size of num[] and rem[].
    // Returns the smallest number
    // x such that:
    // x % num[0] = rem[0],
    // x % num[1] = rem[1],
    // ..................
    // x % num[k-2] = rem[k-1]
    // Assumption: Numbers in num[] 
    // are pairwise coprime (gcd 
    // for every pair is 1)
    static int findMinX(int []num, 
                        int []rem, 
                        int k)
    {
        // Compute product
        // of all numbers
        int prod = 1;
        for (int i = 0; i < k; i++)
            prod *= num[i];
    
        // Initialize result
        int result = 0;
    
        // Apply above formula
        for (int i = 0; i < k; i++)
        {
            int pp = prod / num[i];
            result += rem[i] * 
                      inv(pp, num[i]) * pp;
        }
    
        return result % prod;
    }
    
    // Driver Code
    static public void Main ()
    {
        int []num = {3, 4, 5};
        int []rem = {2, 3, 1};
        int k = num.Length;
        Console.WriteLine("x is " + 
                          findMinX(num, rem, k));
    }
}

// This code is contributed
// by ajit


Output:

x is 11

This article is contributed by Ruchir Garg. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above



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Improved By : Abhinav jain NSIT, jit_t




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