





"""simple docstring"""






from __future__ import annotations

from math import gcd






def      lowercase__ (					lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int = 2							,			lowerCAmelCase__   :			int = 1							,			lowerCAmelCase__   :			int = 3							,			)    ->			int | None:



			'''simple docstring'''
			# A value less than 2 can cause an infinite loop in the algorithm.
			if num < 2:
						raise ValueError("The input value cannot be less than 2"			)

			# Because of the relationship between ``f(f(x))`` and ``f(x)``, this
			# algorithm struggles to find factors that are divisible by two.
			# As a workaround, we specifically check for two and even inputs.
			#   See: https://math.stackexchange.com/a/2856214/165820
			if num > 2 and num % 2 == 0:
						return 2

			# Pollard's Rho algorithm requires a function that returns pseudorandom
			# values between 0 <= X < ``num``.  It doesn't need to be random in the
			# sense that the output value is cryptographically secure or difficult
			# to calculate, it only needs to be random in the sense that all output
			# values should be equally likely to appear.
			# For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
			# However, the success of Pollard's algorithm isn't guaranteed and is
			# determined in part by the initial seed and the chosen random function.
			# To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
			# where ``C`` is a value that we can modify between each attempt.
			def rand_fn(lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int			) -> int:
						return (pow(lowerCAmelCase__							,			2			) + step) % modulus

			for _ in range(lowerCAmelCase__			):
						# These track the position within the cycle detection logic.
						a__  :	Union[str, Any]		 =			seed
						a__  :	Union[str, Any]		 =			seed

						while True:
									# At each iteration, the tortoise moves one step and the hare moves two.
									a__  :	Union[str, Any]		 =			rand_fn(lowerCAmelCase__							,			lowerCAmelCase__							,			lowerCAmelCase__			)
									a__  :	List[Any]		 =			rand_fn(lowerCAmelCase__							,			lowerCAmelCase__							,			lowerCAmelCase__			)
									a__  :	Any		 =			rand_fn(lowerCAmelCase__							,			lowerCAmelCase__							,			lowerCAmelCase__			)

									# At some point both the tortoise and the hare will enter a cycle whose
									# length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
									# the tortoise and hare will end up on the same value modulo ``p``.
									# We can detect when this happens because the position difference between
									# the tortoise and the hare will share a common divisor with ``num``.
									a__  :	Optional[Any]		 =			gcd(hare - tortoise							,			lowerCAmelCase__			)

									if divisor == 1:
												# No common divisor yet, just keep searching.
												continue
									else:
												# We found a common divisor!
												if divisor == num:
															# Unfortunately, the divisor is ``num`` itself and is useless.
															break
												else:
															# The divisor is a nontrivial factor of ``num``!
															return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
						a__  :	str		 =			hare

						# The new step value for the random function can just be incremented.
						# At first the results will be similar to what the old function would
						# have produced, but the value will quickly diverge after a bit.
						step += 1

			# We haven't found a divisor within the requested number of attempts.
			# We were unlucky or ``num`` itself is actually prime.
			return None


if __name__ == "__main__":
		import argparse

		__UpperCAmelCase						=			argparse.ArgumentParser()
		parser.add_argument(
		    '''num''',
		    type=int,
		    help='''The value to find a divisor of''',
		)
		parser.add_argument(
		    '''--attempts''',
		    type=int,
		    default=3,
		    help='''The number of attempts before giving up''',
		)
		__UpperCAmelCase						=			parser.parse_args()

		__UpperCAmelCase						=			pollard_rho(args.num, attempts=args.attempts)
		if divisor is None:
				print(f"{args.num} is probably prime")
		else:
				__UpperCAmelCase						=			args.num // divisor
				print(f"{args.num} = {divisor} * {quotient}")






"""simple docstring"""






from __future__ import annotations






def      lowercase__ (					lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int			)    ->			tuple[int, int]:



			'''simple docstring'''
			if b == 0:
						return (1, 0)
			((a__) ,	(a__))  :	List[Any]		 =			extended_euclid(lowerCAmelCase__							,			a % b			)
			a__  :	str		 =			a // b
			return (y, x - k * y)






def      lowercase__ (					lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int			)    ->			int:



			'''simple docstring'''
			((a__) ,	(a__))  :	Tuple		 =			extended_euclid(lowerCAmelCase__							,			lowerCAmelCase__			)
			a__  :	List[str]		 =			na * na
			a__  :	Union[str, Any]		 =			ra * x * na + ra * y * na
			return (n % m + m) % m






def      lowercase__ (					lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int			)    ->			int:



			'''simple docstring'''
			((a__) ,	(a__))  :	Optional[Any]		 =			extended_euclid(lowerCAmelCase__							,			lowerCAmelCase__			)
			if b < 0:
						a__  :	Optional[int]		 =			(b % n + n) % n
			return b






def      lowercase__ (					lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int							,			lowerCAmelCase__   :			int			)    ->			int:



			'''simple docstring'''
			a__ ,	a__  :	List[Any]		 =			invert_modulo(lowerCAmelCase__							,			lowerCAmelCase__			), invert_modulo(lowerCAmelCase__							,			lowerCAmelCase__			)
			a__  :	Dict		 =			na * na
			a__  :	Any		 =			ra * x * na + ra * y * na
			return (n % m + m) % m


if __name__ == "__main__":
		from doctest import testmod

		testmod(name='''chinese_remainder_theorem''', verbose=True)
		testmod(name='''chinese_remainder_theorem2''', verbose=True)
		testmod(name='''invert_modulo''', verbose=True)
		testmod(name='''extended_euclid''', verbose=True)