Merge pull request #1441 from shubhamamsa/mod_div

feat: added modular division algorithm

DIRECTORY.md

@@ -154,6 +154,7 @@
   * [Least Common Multiple](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/least_common_multiple.cpp)
   * [Magic Number](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/magic_number.cpp)
   * [Miller Rabin](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/miller_rabin.cpp)
+  * [Modular Division](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_division.cpp)
   * [Modular Exponentiation](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_exponentiation.cpp)
   * [Modular Inverse Fermat Little Theorem](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_inverse_fermat_little_theorem.cpp)
   * [N Choose R](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/n_choose_r.cpp)

math/modular_division.cpp

@@ -0,0 +1,115 @@
+/**
+ * @file
+ * @brief An algorithm to divide two numbers under modulo p [Modular
+ * Division](https://www.geeksforgeeks.org/modular-division)
+ * @details To calculate division of two numbers under modulo p
+ * Modulo operator is not distributive under division, therefore
+ * we first have to calculate the inverse of divisor using
+ * [Fermat's little
+ theorem](https://en.wikipedia.org/wiki/Fermat%27s_little_theorem)
+ * Now, we can multiply the dividend with the inverse of divisor
+ * and modulo is distributive over multiplication operation.
+ * Let,
+ * We have 3 numbers a, b, p
+ * To compute (a/b)%p
+ * (a/b)%p ≡ (a*(inverse(b)))%p ≡ ((a%p)*inverse(b)%p)%p
+ * NOTE: For the existence of inverse of 'b', 'b' and 'p' must be coprime
+ * For simplicity we take p as prime
+ * Time Complexity: O(log(b))
+ * Example: ( 24 / 3 ) % 5 => 8 % 5 = 3 --- (i)
+            Now the inverse of 3 is 2
+            (24 * 2) % 5 = (24 % 5) * (2 % 5) = (4 * 2) % 5 = 3 --- (ii)
+            (i) and (ii) are equal hence the answer is correct.
+ * @see modular_inverse_fermat_little_theorem.cpp, modular_exponentiation.cpp
+ * @author [Shubham Yadav](https://github.com/shubhamamsa)
+ */
+
+#include <cassert>   /// for assert
+#include <iostream>  /// for IO operations
+
+/**
+ * @namespace math
+ * @brief Mathematical algorithms
+ */
+namespace math {
+/**
+ * @namespace modular_division
+ * @brief Functions for [Modular
+ * Division](https://www.geeksforgeeks.org/modular-division) implementation
+ */
+namespace modular_division {
+/**
+ * @brief This function calculates a raised to exponent b under modulo c using
+ * modular exponentiation.
+ * @param a integer base
+ * @param b unsigned integer exponent
+ * @param c integer modulo
+ * @return a raised to power b modulo c
+ */
+uint64_t power(uint64_t a, uint64_t b, uint64_t c) {
+    uint64_t ans = 1;  /// Initialize the answer to be returned
+    a = a % c;         /// Update a if it is more than or equal to c
+    if (a == 0) {
+        return 0;  /// In case a is divisible by c;
+    }
+    while (b > 0) {
+        /// If b is odd, multiply a with answer
+        if (b & 1) {
+            ans = ((ans % c) * (a % c)) % c;
+        }
+        /// b must be even now
+        b = b >> 1;  /// b = b/2
+        a = ((a % c) * (a % c)) % c;
+    }
+    return ans;
+}
+
+/**
+ * @brief This function calculates modular division
+ * @param a integer dividend
+ * @param b integer divisor
+ * @param p integer modulo
+ * @return a/b modulo c
+ */
+uint64_t mod_division(uint64_t a, uint64_t b, uint64_t p) {
+    uint64_t inverse = power(b, p - 2, p) % p;  /// Calculate the inverse of b
+    uint64_t result =
+        ((a % p) * (inverse % p)) % p;  /// Calculate the final result
+    return result;
+}
+}  // namespace modular_division
+}  // namespace math
+
+/**
+ * Function for testing power function.
+ * test cases and assert statement.
+ * @returns `void`
+ */
+static void test() {
+    uint64_t test_case_1 = math::modular_division::mod_division(8, 2, 2);
+    assert(test_case_1 == 0);
+    std::cout << "Test 1 Passed!" << std::endl;
+    uint64_t test_case_2 = math::modular_division::mod_division(15, 3, 7);
+    assert(test_case_2 == 5);
+    std::cout << "Test 2 Passed!" << std::endl;
+    uint64_t test_case_3 = math::modular_division::mod_division(10, 5, 2);
+    assert(test_case_3 == 0);
+    std::cout << "Test 3 Passed!" << std::endl;
+    uint64_t test_case_4 = math::modular_division::mod_division(81, 3, 5);
+    assert(test_case_4 == 2);
+    std::cout << "Test 4 Passed!" << std::endl;
+    uint64_t test_case_5 = math::modular_division::mod_division(12848, 73, 29);
+    assert(test_case_5 == 2);
+    std::cout << "Test 5 Passed!" << std::endl;
+}
+
+/**
+ * @brief Main function
+ * @param argc commandline argument count (ignored)
+ * @param argv commandline array of arguments (ignored)
+ * @returns 0 on exit
+ */
+int main(int argc, char *argv[]) {
+    test();  // execute the tests
+    return 0;
+}