Add chinese remainder theorem (#5873)

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

@@ -0,0 +1,84 @@
+package com.thealgorithms.maths;
+
+import java.util.List;
+
+/**
+ * @brief Implementation of the Chinese Remainder Theorem (CRT) algorithm
+ * @details
+ * The Chinese Remainder Theorem (CRT) is used to solve systems of
+ * simultaneous congruences. Given several pairwise coprime moduli
+ * and corresponding remainders, the algorithm finds the smallest
+ * positive solution.
+ */
+public final class ChineseRemainderTheorem {
+    private ChineseRemainderTheorem() {
+    }
+
+    /**
+     * @brief Solves the Chinese Remainder Theorem problem.
+     * @param remainders The list of remainders.
+     * @param moduli The list of pairwise coprime moduli.
+     * @return The smallest positive solution that satisfies all the given congruences.
+     */
+    public static int solveCRT(List<Integer> remainders, List<Integer> moduli) {
+        int product = 1;
+        int result = 0;
+
+        // Calculate the product of all moduli
+        for (int mod : moduli) {
+            product *= mod;
+        }
+
+        // Apply the formula for each congruence
+        for (int i = 0; i < moduli.size(); i++) {
+            int partialProduct = product / moduli.get(i);
+            int inverse = modInverse(partialProduct, moduli.get(i));
+            result += remainders.get(i) * partialProduct * inverse;
+        }
+
+        // Adjust result to be the smallest positive solution
+        result = result % product;
+        if (result < 0) {
+            result += product;
+        }
+
+        return result;
+    }
+
+    /**
+     * @brief Computes the modular inverse of a number with respect to a modulus using
+     * the Extended Euclidean Algorithm.
+     * @param a The number for which to find the inverse.
+     * @param m The modulus.
+     * @return The modular inverse of a modulo m.
+     */
+    private static int modInverse(int a, int m) {
+        int m0 = m;
+        int x0 = 0;
+        int x1 = 1;
+
+        if (m == 1) {
+            return 0;
+        }
+
+        while (a > 1) {
+            int q = a / m;
+            int t = m;
+
+            // m is remainder now, process same as Euclid's algorithm
+            m = a % m;
+            a = t;
+            t = x0;
+
+            x0 = x1 - q * x0;
+            x1 = t;
+        }
+
+        // Make x1 positive
+        if (x1 < 0) {
+            x1 += m0;
+        }
+
+        return x1;
+    }
+}