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"""Implementation of Basic Math in Python."""

import math

def prime_factors(n: int) -> list:

"""Find Prime Factors.

>>> prime_factors(100)

[2, 2, 5, 5]

"""

pf = []

while n % 2 == 0:

pf.append(2)

n = int(n / 2)

for i in range(3, int(math.sqrt(n)) + 1, 2):

while n % i == 0:

pf.append(i)

n = int(n / i)

if n > 2:

pf.append(n)

return pf

def number_of_divisors(n: int) -> int:

"""Calculate Number of Divisors of an Integer.

>>> number_of_divisors(100)

9

"""

div = 1

temp = 1

while n % 2 == 0:

temp += 1

n = int(n / 2)

div *= temp

for i in range(3, int(math.sqrt(n)) + 1, 2):

temp = 1

while n % i == 0:

temp += 1

n = int(n / i)

div *= temp

return div

def sum_of_divisors(n: int) -> int:

"""Calculate Sum of Divisors.

>>> sum_of_divisors(100)

217

"""

s = 1

temp = 1

while n % 2 == 0:

temp += 1

n = int(n / 2)

if temp > 1:

s *= (2 ** temp - 1) / (2 - 1)

for i in range(3, int(math.sqrt(n)) + 1, 2):

temp = 1

while n % i == 0:

temp += 1

n = int(n / i)

if temp > 1:

s *= (i ** temp - 1) / (i - 1)

return int(s)

def euler_phi(n: int) -> int:

"""Calculate Euler's Phi Function.

>>> euler_phi(100)

40

"""

s = n

for x in set(prime_factors(n)):

s *= (x - 1) / x

return int(s)

if __name__ == "__main__":

print(prime_factors(100))

print(number_of_divisors(100))

print(sum_of_divisors(100))

print(euler_phi(100))