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Chinese Remainder Theorem | Set 2 (Inverse Modulo based Implementation)
  • Difficulty Level : Hard
  • Last Updated : 05 Apr, 2021

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
     = (80 + 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:
// 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:
    // 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 Python3 program 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:
# 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:
    // 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

PHP




<?php
// PHP program 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:
// multiplicative-inverse-under-modulo-m/
function inv($a, $m)
{
    $m0 = $m;
    $x0 = 0;
    $x1 = 1;
 
    if ($m == 1)
    return 0;
 
    // Apply extended Euclid Algorithm
    while ($a > 1)
    {
        // q is quotient
        $q = (int)($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)
function findMinX($num, $rem, $k)
{
    // Compute product of all numbers
    $prod = 1;
    for ($i = 0; $i < $k; $i++)
        $prod *= $num[$i];
 
    // Initialize result
    $result = 0;
 
    // Apply above formula
    for ($i = 0; $i < $k; $i++)
    {
        $pp = (int)$prod / $num[$i];
        $result += $rem[$i] * inv($pp,
                    $num[$i]) * $pp;
    }
 
    return $result % $prod;
}
 
// Driver Code
$num = array(3, 4, 5);
$rem = array(2, 3, 1);
$k = sizeof($num);
echo "x is ". findMinX($num, $rem, $k);
 
// This code is contributed by mits
?>

Javascript




<script>
// Javascript program 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:
// multiplicative-inverse-under-modulo-m/
function inv(a, m)
{
    let m0 = m;
    let x0 = 0;
    let x1 = 1;
 
    if (m == 1)
    return 0;
 
    // Apply extended Euclid Algorithm
    while (a > 1)
    {
        // q is quotient
        let q = parseInt(a / m);
 
        let 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)
function findMinX(num, rem, k)
{
    // Compute product of all numbers
    let prod = 1;
    for (let i = 0; i < k; i++)
        prod *= num[i];
 
    // Initialize result
    let result = 0;
 
    // Apply above formula
    for (let i = 0; i < k; i++)
    {
        pp = parseInt(prod / num[i]);
        result += rem[i] * inv(pp,
                    num[i]) * pp;
    }
 
    return result % prod;
}
 
// Driver Code
let num = new Array(3, 4, 5);
let rem = new Array(2, 3, 1);
let k = num.length;
document.write("x is " + findMinX(num, rem, k));
 
// This code is contributed by _saurabh_jaiswal
</script>

Output: 

x is 11

Time Complexity : O(N*LogN)

Auxiliary Space : O(1)

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 

Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the Essential Maths for CP Course at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

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