Added Chines Remainder Theorem

Author: volimroze

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

@@ -0,0 +1,71 @@
+package remainder;
+
+import java.util.*;
+
+public class ChineseRemainderTheorem {
+
+    // Helper function to compute the greatest common divisor (GCD)
+    public static int gcd(int a, int b) {
+        if (b == 0) return a;
+        return gcd(b, a % b);
+    }
+
+    // Helper function to compute the modular inverse of a modulo m
+    // Uses the extended Euclidean algorithm
+    public static int modInverse(int a, int m) {
+        int m0 = m, t, q;
+        int x0 = 0, x1 = 1;
+
+        if (m == 1) return 0;
+
+        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;
+    }
+
+    // Function to find the smallest x that satisfies
+    // the given system of congruences using the Chinese Remainder Theorem
+    public 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 the formula:
+        // result = (a1 * N1 * m1 + a2 * N2 * m2 + ... + ak * Nk * mk) % prod
+        for (int i = 0; i < k; i++) {
+            int pp = prod / num[i]; // Compute Ni = N / ni
+            result += rem[i] * pp * modInverse(pp, num[i]);
+        }
+
+        return result % prod;
+    }
+
+    public static void main(String[] args) {
+        // Example case
+        int[] num = {3, 5, 7}; // Moduli
+        int[] rem = {2, 3, 2}; // Remainders
+        int k = num.length;
+
+        // Calculate and print the result
+        int result = findMinX(num, rem, k);
+        System.out.println("The smallest x that satisfies the given system of congruences is: " + result);
+    }
+}