Understanding mathematics and algorithm problems is very important when preparing for coding tests. Today, we will discuss the Euler’s phi function and how to implement it in C++. The Euler’s phi function returns the number of positive integers less than or equal to a given integer The Euler’s phi function is defined as follows:<\/p>\n where p1, p2, … pk are all the prime numbers that are coprime to n. For example, \u03c6(6) is calculated as follows:<\/p>\n Problem: Write a C++ program that calculates the Euler’s phi function for a given integer Input: An integer Output: An integer \u03c6(n)<\/p>\n Calculating the Euler’s phi function directly may be inefficient. We can precompute the primes and then use the formula defined above to calculate \u03c6(n). To do this, we design the algorithm in the following steps.<\/p>\n The Sieve of Eratosthenes is a classical algorithm for quickly finding primes. This algorithm is highly efficient with a time complexity of O(n log log n).<\/p>\n Now, let’s implement the complete algorithm in C++. Below is the C++ code that calculates the Euler’s phi function.<\/p>\n The above code works in the following steps:<\/p>\n After completing the code, we can validate the function with various test cases. For example:<\/p>\nn<\/code> that are coprime to n<\/code>. It has many applications in number theory and is useful for solving mathematical problems.<\/p>\n1. Introduction to Euler’s Phi Function<\/h2>\n
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1.1 Properties of Euler’s Phi Function<\/h3>\n
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2. Problem Description<\/h2>\n
n<\/code>.<\/p>\nn<\/code> (1 \u2264 n \u2264 106<\/sup>)<\/p>\n3. Algorithm Design<\/h2>\n
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3.1 Sieve of Eratosthenes<\/h3>\n
4. C++ Code Implementation<\/h2>\n
#include <iostream>\n#include <vector>\n\nusing namespace std;\n\n\/\/ Calculate Euler's phi function\nint eulerPhi(int n) {\n int result = n; \/\/ Initial value is n\n for (int i = 2; i * i <= n; i++) {\n \/\/ If i is a divisor of n\n if (n % i == 0) {\n \/\/ Adjust the result by multiplying the prime\n while (n % i == 0) {\n n \/= i;\n }\n result -= result \/ i; \/\/ Calculate the result\n }\n }\n \/\/ If the remaining n is a prime\n if (n > 1) {\n result -= result \/ n; \/\/ Process the result for the prime\n }\n return result;\n}\n\nint main() {\n int n;\n cout << \"Enter an integer: \";\n cin >> n;\n \n cout << \"\u03c6(\" << n << \") = \" << eulerPhi(n) << endl;\n return 0;\n}\n<\/code><\/pre>\n5. Code Explanation<\/h2>\n
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eulerPhi(int n)<\/code> function calculates and returns the Euler’s phi function for the given integer n.<\/li>\nresult<\/code> initially holds the value of n and is updated whenever a prime is found.<\/li>\ni<\/code> is a divisor of n, it removes the multiples of i by dividing n by i. During this process, it adjusts the result according to the prime.<\/li>\n6. Code Testing<\/h2>\n