



"""simple docstring"""







from __future__ import annotations

from math import gcd

def   lowerCAmelCase__			(     lowerCamelCase__				,      lowerCamelCase__ = 2				,      lowerCamelCase__ = 1				,      lowerCamelCase__ = 3				,      ) ->							int | None:
							# 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__				,      lowerCamelCase__				,      lowerCamelCase__			) -> int:
														return (pow(lowerCamelCase__				,      2			) + step) % modulus

							for _ in range(lowerCamelCase__			):
														# These track the position within the cycle detection logic.
														A         =					seed
														A         =					seed

														while True:
																					# At each iteration, the tortoise moves one step and the hare moves two.
																					A         =					rand_fn(lowerCamelCase__				,      lowerCamelCase__				,      lowerCamelCase__			)
																					A         =					rand_fn(lowerCamelCase__				,      lowerCamelCase__				,      lowerCamelCase__			)
																					A         =					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         =					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         =					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

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

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








"""simple docstring"""







def   lowerCAmelCase__			(     lowerCamelCase__			) ->							int:

							if not isinstance(lowerCamelCase__				,      lowerCamelCase__			):
														raise ValueError('multiplicative_persistence() only accepts integral values'			)
							if num < 0:
														raise ValueError('multiplicative_persistence() does not accept negative values'			)

							A         =					0
							A         =					str(lowerCamelCase__			)

							while len(lowerCamelCase__			) != 1:
														A         =					[int(lowerCamelCase__			) for i in num_string]

														A         =					1
														for i in range(0				,      len(lowerCamelCase__			)			):
																					total *= numbers[i]

														A         =					str(lowerCamelCase__			)

														steps += 1
							return steps

def   lowerCAmelCase__			(     lowerCamelCase__			) ->							int:

							if not isinstance(lowerCamelCase__				,      lowerCamelCase__			):
														raise ValueError('additive_persistence() only accepts integral values'			)
							if num < 0:
														raise ValueError('additive_persistence() does not accept negative values'			)

							A         =					0
							A         =					str(lowerCamelCase__			)

							while len(lowerCamelCase__			) != 1:
														A         =					[int(lowerCamelCase__			) for i in num_string]

														A         =					0
														for i in range(0				,      len(lowerCamelCase__			)			):
																					total += numbers[i]

														A         =					str(lowerCamelCase__			)

														steps += 1
							return steps


if __name__ == "__main__":
							import doctest

							doctest.testmod()



