basic_maths.py

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

import math


def gcd(a: int, b: int) -> int:
    """Calculate the Greatest Common Divisor (GCD) using Euclid's Algorithm.
    >>> gcd(54, 24)
    6
    >>> gcd(10, 0)
    10
    >>> gcd(0, 10)
    10
    >>> gcd(0, 0)
    Traceback (most recent call last):
        ...
    ValueError: At least one number must be non-zero
    >>> gcd(-54, 24)
    6
    """
    if a == 0 and b == 0:
        raise ValueError("At least one number must be non-zero")
    a, b = abs(a), abs(b)
    while b:
        a, b = b, a % b
    return a


def lcm(a: int, b: int) -> int:
    """Calculate the Least Common Multiple (LCM) using GCD.
    >>> lcm(12, 15)
    60
    >>> lcm(0, 10)
    0
    >>> lcm(-12, 15)
    60
    """
    if a == 0 or b == 0:
        return 0
    return abs(a * b) // gcd(a, b)


def prime_factors(n: int) -> list:
    """
    Uses a standard method of dividing by 2, then odd divisors up to sqrt(n).
    Time Complexity: O(sqrt(n))
    Space Complexity: O(log n) on average
    """
    """Find Prime Factors.
    >>> prime_factors(100)
    [2, 2, 5, 5]
    >>> prime_factors(0)
    Traceback (most recent call last):
        ...
    ValueError: Only positive integers have prime factors
    >>> prime_factors(-10)
    Traceback (most recent call last):
        ...
    ValueError: Only positive integers have prime factors
    """
    if n <= 0:
        raise ValueError("Only positive integers have prime factors")
    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
    >>> number_of_divisors(0)
    Traceback (most recent call last):
        ...
    ValueError: Only positive numbers are accepted
    >>> number_of_divisors(-10)
    Traceback (most recent call last):
        ...
    ValueError: Only positive numbers are accepted
    """
    if n <= 0:
        raise ValueError("Only positive numbers are accepted")
    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
    if n > 1:
        div *= 2
    return div


def sum_of_divisors(n: int) -> int:
    """Calculate Sum of Divisors.
    >>> sum_of_divisors(100)
    217
    >>> sum_of_divisors(0)
    Traceback (most recent call last):
        ...
    ValueError: Only positive numbers are accepted
    >>> sum_of_divisors(-10)
    Traceback (most recent call last):
        ...
    ValueError: Only positive numbers are accepted
    """
    if n <= 0:
        raise ValueError("Only positive numbers are accepted")
    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
    >>> euler_phi(0)
    Traceback (most recent call last):
        ...
    ValueError: Only positive numbers are accepted
    >>> euler_phi(-10)
    Traceback (most recent call last):
        ...
    ValueError: Only positive numbers are accepted
    """
    if n <= 0:
        raise ValueError("Only positive numbers are accepted")
    s = n
    for x in set(prime_factors(n)):
        s *= (x - 1) / x
    return int(s)


if __name__ == "__main__":
    import doctest

    doctest.testmod()