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"""

Extended Euclidean Algorithm.

Finds 2 numbers a and b such that it satisfies

the equation am + bn = gcd(m, n) (a.k.a Bezout's Identity)

https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm

"""

# @Author: S. Sharma <silentcat>

# @Date: 2019-02-25T12:08:53-06:00

# @Email: silentcat@protonmail.com

# @Last modified by: pikulet

# @Last modified time: 2020-10-02

from __future__ import annotations

import sys

def extended_euclidean_algorithm(a: int, b: int) -> tuple[int, int]:

"""

Extended Euclidean Algorithm.

Finds 2 numbers a and b such that it satisfies

the equation am + bn = gcd(m, n) (a.k.a Bezout's Identity)

>>> extended_euclidean_algorithm(1, 24)

(1, 0)

>>> extended_euclidean_algorithm(8, 14)

(2, -1)

>>> extended_euclidean_algorithm(240, 46)

(-9, 47)

>>> extended_euclidean_algorithm(1, -4)

(1, 0)

>>> extended_euclidean_algorithm(-2, -4)

(-1, 0)

>>> extended_euclidean_algorithm(0, -4)

(0, -1)

>>> extended_euclidean_algorithm(2, 0)

(1, 0)

"""

# base cases

if abs(a) == 1:

return a, 0

elif abs(b) == 1:

return 0, b

old_remainder, remainder = a, b

old_coeff_a, coeff_a = 1, 0

old_coeff_b, coeff_b = 0, 1

while remainder != 0:

quotient = old_remainder // remainder

old_remainder, remainder = remainder, old_remainder - quotient * remainder

old_coeff_a, coeff_a = coeff_a, old_coeff_a - quotient * coeff_a

old_coeff_b, coeff_b = coeff_b, old_coeff_b - quotient * coeff_b

# sign correction for negative numbers

if a < 0:

old_coeff_a = -old_coeff_a

if b < 0:

old_coeff_b = -old_coeff_b

return old_coeff_a, old_coeff_b

def main():

"""Call Extended Euclidean Algorithm."""

if len(sys.argv) < 3:

print("2 integer arguments required")

exit(1)

a = int(sys.argv[1])

b = int(sys.argv[2])

print(extended_euclidean_algorithm(a, b))

if __name__ == "__main__":

main()

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