feat: largest_power (#1406)

* largest_power

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update math/largest_power.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* added a new line

* updating DIRECTORY.md

Co-authored-by: David Leal <halfpacho@gmail.com>
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

DIRECTORY.md

@@ -148,6 +148,7 @@
   * [Gcd Recursive Euclidean](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/gcd_recursive_euclidean.cpp)
   * [Large Factorial](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/large_factorial.cpp)
   * [Large Number](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/large_number.h)
+  * [Largest Power](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/largest_power.cpp)
   * [Lcm Sum](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/lcm_sum.cpp)
   * [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)

math/largest_power.cpp

@@ -0,0 +1,78 @@
+/**
+ * @file
+ * @brief Algorithm to find largest x such that p^x divides n! (factorial) using Legendre's Formula.
+ * @details Given an integer n and a prime number p, the task is to find the largest x such that 
+ * p^x (p raised to power x) divides n! (factorial). This will be done using Legendre's formula:
+ * x = [n/(p^1)] + [n/(p^2)] + [n/(p^3)] + \ldots + 1
+ * @see more on https://math.stackexchange.com/questions/141196/highest-power-of-a-prime-p-dividing-n
+ * @author [uday6670](https://github.com/uday6670)
+ */
+
+#include <iostream> /// for std::cin and std::cout
+#include <cassert>  /// for assert
+
+/**
+ * @namespace math
+ * @brief Mathematical algorithms
+ */
+namespace math {
+
+	/** 
+	 * @brief Function to calculate largest power
+	 * @param n number
+         * @param p prime number
+	 * @returns largest power
+	 */ 
+    uint64_t largestPower(uint32_t n, const uint16_t& p)  
+    {  
+        // Initialize result  
+        int x = 0;  
+  
+        // Calculate result 
+        while (n)  
+        {  
+            n /= p;  
+            x += n;  
+        }  
+        return x;  
+    }
+
+}  // namespace math
+
+/**
+ * @brief Function for testing largestPower function.
+ * test cases and assert statement.
+ * @returns `void`
+*/
+static void test()
+{
+	uint8_t test_case_1 = math::largestPower(5,2);
+	assert(test_case_1==3);
+	std::cout<<"Test 1 Passed!"<<std::endl;
+	
+	uint16_t test_case_2 = math::largestPower(10,3);
+	assert(test_case_2==4);
+	std::cout<<"Test 2 Passed!"<<std::endl;
+	
+	uint32_t test_case_3 = math::largestPower(25,5);
+	assert(test_case_3==6);
+	std::cout<<"Test 3 Passed!"<<std::endl;
+	
+	uint32_t test_case_4 = math::largestPower(27,2);
+	assert(test_case_4==23);
+	std::cout<<"Test 4 Passed!"<<std::endl;
+	
+	uint16_t test_case_5 = math::largestPower(7,3);
+	assert(test_case_5==2);
+	std::cout<<"Test 5 Passed!"<<std::endl;
+} 
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() 
+{ 
+    test(); // execute the tests
+    return 0;
+}