






"""simple docstring"""
from __future__ import annotations

from math import gcd

def 				snake_case      (   A__	,A__ = 2	,A__ = 1	,A__ = 3	,):
			# 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(A__	,A__	,A__						) -> int:
						return (pow(A__	,2						) + step) % modulus

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

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

									# 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``.
									UpperCAmelCase_     : Optional[int]			     =					gcd(hare - tortoise	,A__						)

									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.
						UpperCAmelCase_     : List[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

			lowerCamelCase_					=      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''',
			)
			lowerCamelCase_					=      parser.parse_args()

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











"""simple docstring"""
lowerCamelCase_					=      [
    (1000, '''M'''),
    (900, '''CM'''),
    (500, '''D'''),
    (400, '''CD'''),
    (100, '''C'''),
    (90, '''XC'''),
    (50, '''L'''),
    (40, '''XL'''),
    (10, '''X'''),
    (9, '''IX'''),
    (5, '''V'''),
    (4, '''IV'''),
    (1, '''I'''),
]

def 				snake_case      (   A__						):
			UpperCAmelCase_     : List[str]			     =					{"I": 1, "V": 5, "X": 10, "L": 50, "C": 1_00, "D": 5_00, "M": 10_00}
			UpperCAmelCase_     : Optional[Any]			     =					0
			UpperCAmelCase_     : Tuple			     =					0
			while place < len(A__						):
						if (place + 1 < len(A__						)) and (vals[roman[place]] < vals[roman[place + 1]]):
									total += vals[roman[place + 1]] - vals[roman[place]]
									place += 2
						else:
									total += vals[roman[place]]
									place += 1
			return total

def 				snake_case      (   A__						):
			UpperCAmelCase_     : Union[str, Any]			     =					[]
			for arabic, roman in ROMAN:
						((UpperCAmelCase_)     ,	(UpperCAmelCase_))     : str			     =					divmod(A__	,A__						)
						result.append(roman * factor						)
						if number == 0:
									break
			return "".join(A__						)


if __name__ == "__main__":
			import doctest

			doctest.testmod()



