diff --git a/DIRECTORY.md b/DIRECTORY.md index cd8f6fb85..71da6a402 100644 --- a/DIRECTORY.md +++ b/DIRECTORY.md @@ -697,6 +697,8 @@ * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_067/sol1.py) * Problem 069 * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_069/sol1.py) + * Problem 070 + * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_070/sol1.py) * Problem 071 * [Sol1](https://github.com/TheAlgorithms/Python/blob/master/project_euler/problem_071/sol1.py) * Problem 072 diff --git a/project_euler/problem_070/__init__.py b/project_euler/problem_070/__init__.py new file mode 100644 index 000000000..e69de29bb diff --git a/project_euler/problem_070/sol1.py b/project_euler/problem_070/sol1.py new file mode 100644 index 000000000..9d27119ba --- /dev/null +++ b/project_euler/problem_070/sol1.py @@ -0,0 +1,119 @@ +""" +Project Euler Problem 70: https://projecteuler.net/problem=70 + +Euler's Totient function, φ(n) [sometimes called the phi function], is used to +determine the number of positive numbers less than or equal to n which are +relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than +nine and relatively prime to nine, φ(9)=6. + +The number 1 is considered to be relatively prime to every positive number, so +φ(1)=1. + +Interestingly, φ(87109)=79180, and it can be seen that 87109 is a permutation +of 79180. + +Find the value of n, 1 < n < 10^7, for which φ(n) is a permutation of n and +the ratio n/φ(n) produces a minimum. + +----- + +This is essentially brute force. Calculate all totients up to 10^7 and +find the minimum ratio of n/φ(n) that way. To minimize the ratio, we want +to minimize n and maximize φ(n) as much as possible, so we can store the +minimum fraction's numerator and denominator and calculate new fractions +with each totient to compare against. To avoid dividing by zero, I opt to +use cross multiplication. + +References: +Finding totients +https://en.wikipedia.org/wiki/Euler's_totient_function#Euler's_product_formula +""" +from typing import List + + +def get_totients(max_one: int) -> List[int]: + """ + Calculates a list of totients from 0 to max_one exclusive, using the + definition of Euler's product formula. + + >>> get_totients(5) + [0, 1, 1, 2, 2] + + >>> get_totients(10) + [0, 1, 1, 2, 2, 4, 2, 6, 4, 6] + """ + totients = [0] * max_one + + for i in range(0, max_one): + totients[i] = i + + for i in range(2, max_one): + if totients[i] == i: + for j in range(i, max_one, i): + totients[j] -= totients[j] // i + + return totients + + +def has_same_digits(num1: int, num2: int) -> bool: + """ + Return True if num1 and num2 have the same frequency of every digit, False + otherwise. + + digits[] is a frequency table where the index represents the digit from + 0-9, and the element stores the number of appearances. Increment the + respective index every time you see the digit in num1, and decrement if in + num2. At the end, if the numbers have the same digits, every index must + contain 0. + + >>> has_same_digits(123456789, 987654321) + True + + >>> has_same_digits(123, 12) + False + + >>> has_same_digits(1234566, 123456) + False + """ + digits = [0] * 10 + + while num1 > 0 and num2 > 0: + digits[num1 % 10] += 1 + digits[num2 % 10] -= 1 + num1 //= 10 + num2 //= 10 + + for digit in digits: + if digit != 0: + return False + + return True + + +def solution(max: int = 10000000) -> int: + """ + Finds the value of n from 1 to max such that n/φ(n) produces a minimum. + + >>> solution(100) + 21 + + >>> solution(10000) + 4435 + """ + + min_numerator = 1 # i + min_denominator = 0 # φ(i) + totients = get_totients(max + 1) + + for i in range(2, max + 1): + t = totients[i] + + if i * min_denominator < min_numerator * t and has_same_digits(i, t): + min_numerator = i + min_denominator = t + + return min_numerator + + +if __name__ == "__main__": + print(f"{solution() = }")