{"id":33714,"date":"2024-11-01T09:19:33","date_gmt":"2024-11-01T09:19:33","guid":{"rendered":"http:\/\/atmokpo.com\/w\/?p=33714"},"modified":"2024-11-01T11:47:00","modified_gmt":"2024-11-01T11:47:00","slug":"python-coding-test-course-euler-pi","status":"publish","type":"post","link":"https:\/\/atmokpo.com\/w\/33714\/","title":{"rendered":"python coding test course, Euler pi"},"content":{"rendered":"

<\/p>\n

\n<\/header>\n
\n

Problem Statement<\/h2>\n

\n Euler’s totient function \u03c6(n) counts the number of integers
\n from 1 to n that are coprime to n. For example,
\n \u03c6(1)=1, \u03c6(2)=1, \u03c6(3)=2, \u03c6(4)=2, \u03c6(5)=4.\n <\/p>\n

\n Write a code to calculate the value of \u03c6(n) for a given positive integer n.
\n Note that n is a value between 1 and 10^6.\n <\/p>\n

Problem Solving Process<\/h2>\n

1. Understanding the Problem<\/h3>\n

\n This problem is based on the definition of Euler’s totient function,
\n and we need to find all the numbers that are coprime to n. Two numbers are
\n said to be coprime if their greatest common divisor is 1, and using this
\n condition we can calculate \u03c6(n).\n <\/p>\n

2. Algorithm Design<\/h3>\n

\n The Euler’s totient function can be expressed in the form of irreducible fractions. By this, we can find
\n the prime factors of n and calculate the number of coprime integers. For a prime factor p of n,
\n \u03c6(n) = n \u00d7 (1 – 1\/p). We need to apply this formula for all prime factors of n.\n <\/p>\n

3. Code Implementation<\/h3>\n

\n First, I will implement a function to find the prime factors, and then use it to
\n write the code to calculate \u03c6(n).\n <\/p>\n

\ndef euler_totient(n):\n    result = n   # Initial value is n\n    p = 2\n    while p * p <= n:   # Finding prime factors\n        if n % p == 0:   # Check if p is a prime factor of n\n            while n % p == 0:   # Remove the prime factors corresponding to p\n                n \/\/= p\n            result -= result \/\/ p   # Calculate the number of coprimes\n        p += 1\n    if n > 1:   # The last remaining prime factor\n        result -= result \/\/ n\n    return result\n        <\/code><\/pre>\n
4. Examples and Tests<\/h3>\n
\n            Let’s calculate \u03c6(10). \u03c6(10) has the prime factors 5 and 2.
\n            Thus, \u03c6(10) = 10 \u00d7 (1 – 1\/2) \u00d7 (1 – 1\/5) = 4.\n        <\/p>\n
\nprint(euler_totient(10))  # Output: 4\n        <\/code><\/pre>\n
5. Conclusion<\/h3>\n
\n            Euler’s totient function is a very important concept in number theory,
\n            providing essential knowledge for implementing advanced algorithms.
\n            This problem helps establish this foundational knowledge.\n        <\/p>\n<\/article>\n