project_euler.problem_072.sol1

Problem 72 Counting fractions: https://projecteuler.net/problem=72

Description:

Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction. If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we get: 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8 It can be seen that there are 21 elements in this set. How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000?

Solution:

Number of numbers between 1 and n that are coprime to n is given by the Euler’s Totient function, phi(n). So, the answer is simply the sum of phi(n) for 2 <= n <= 1,000,000 Sum of phi(d), for all d|n = n. This result can be used to find phi(n) using a sieve.

Time: 1 sec

Functions

solution(→ int)

Returns an integer, the solution to the problem

Module Contents

project_euler.problem_072.sol1.solution(limit: int = 1000000) → int

Returns an integer, the solution to the problem >>> solution(10) 31 >>> solution(100) 3043 >>> solution(1_000) 304191