Chinese Remainder Theorem Implementation

This commit implements the Chinese Remainder Theorem (CRT) algorithm in Java and adds detailed comments to explain the functionality of each part of the code. It includes functions for Greatest Common Divisor (GCD) calculation, modular inverse computation, and the main CRT logic.

Author: mn121mn121

src/main/java/com/thealgorithms/maths/ChineseRemainderTheorem.java

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+//this program is to implement Chinese Remainder Theorem
+public class ChineseRemainderTheorem
+{
+
+    //function to find the greatest common divisor (GCD) of two numbers using the Extended Euclidean Algorithm
+    private static long gcd(long a, long b)
+	{
+        if(b == 0)
+            return a; //base case: if b is 0, a is the GCD
+
+        return gcd(b, a%b);//calling recursively
+    }
+
+    //function to find the modular inverse of 'a' under modulo 'm' using the Extended Euclidean Algorithm
+    private static long extendedGCD(long a, long b)
+	{
+        long originalB = b; //keeping original 'b' for later use
+        long x1 = 1, x2 = 0; //x1 and x2 are coefficients for 'a'
+        long y1 = 0, y2 = 1; //y1 and y2 are coefficients for 'b'
+
+        while(b != 0)
+		{
+            long q = a/b; //quotient
+            long temp = b; //temporary variable to hold value of 'b'
+            b = a % b; //remainder
+            a = temp; //updating 'a' with previous value of 'b'
+
+            //updating coefficients x1, x2, y1, y2
+            temp = x2;
+            x2 = x1 - q*x2; //updating x2 with the quotient
+            x1 = temp;
+
+            temp = y2;
+            y2 = y1 - q*y2; //updating y2 with the quotient
+            y1 = temp;
+        }
+        //ensuring the result is positive
+        return (x1 < 0)? (x1+originalB) : x1;
+    }
+
+    //function to implement the Chinese Remainder Theorem
+    public static long chineseRemainder(int[] n, int[] a)
+	{
+        long N = 1; //variable to hold the product of all moduli
+        for(int ni : n)
+            N *= ni; //calculating the product of all moduli
+
+        long result = 0; //variable to accumulate the final result
+        for (int i = 0; i < n.length; i++)
+		{
+            long Ni = N/n[i]; //calculating Ni (the product of all moduli except n[i])
+            long mi = extendedGCD(Ni, n[i]); //finding the modular inverse of Ni modulo n[i]
+            //updating the result
+            result += a[i]*Ni*mi; 
+        }
+
+        return result%N; //returning final answer
+    }
+
+    public static void main(String[] args)
+	{
+        //example test case
+        int[] n = {3, 5, 7}; //moduli
+        int[] a = {2, 3, 2}; //remainders
+
+        //calculating the solution using the Chinese Remainder Theorem
+        long solution = chineseRemainder(n, a);
+        //output
+        System.out.println("The solution is: " + solution); //expected output: 23
+    }
+}