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-1rem[i]is given array of remaindersprodis 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