

'''simple docstring'''


from __future__ import annotations

from math import gcd




def 	_a       (				__lowerCAmelCase		:      int   ,							__lowerCAmelCase		:      int = 2   ,							__lowerCAmelCase		:      int = 1   ,							__lowerCAmelCase		:      int = 3   ,							):

  """simple docstring"""

  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.
    snake_case__						:       List[str]       						=  seed
    snake_case__						:       Union[str, Any]       						=  seed

    while True:
      # At each iteration, the tortoise moves one step and the hare moves two.
      snake_case__						:       Dict       						=  rand_fn(__lowerCAmelCase   ,							__lowerCAmelCase   ,							__lowerCAmelCase     )
      snake_case__						:       Optional[int]       						=  rand_fn(__lowerCAmelCase   ,							__lowerCAmelCase   ,							__lowerCAmelCase     )
      snake_case__						:       List[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``.
      snake_case__						:       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.
    snake_case__						:       List[Any]       						=  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__						:       Dict										=						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__						:       Union[str, Any]										=						parser.parse_args()

     lowerCAmelCase__						:       Tuple										=						pollard_rho(args.num, attempts=args.attempts)
     if divisor is None:
          print(f"""{args.num} is probably prime""")
     else:
          lowerCAmelCase__						:       Optional[Any]										=						args.num // divisor
          print(f"""{args.num} = {divisor} * {quotient}""")






'''simple docstring'''


def 	_a       (				__lowerCAmelCase		:      float   ,							__lowerCAmelCase		:      float   ,							__lowerCAmelCase		:      int     ):

		"""simple docstring"""

		if principal <= 0:
				raise Exception('''Principal borrowed must be > 0'''     )
		if rate_per_annum < 0:
				raise Exception('''Rate of interest must be >= 0'''     )
		if years_to_repay <= 0 or not isinstance(__lowerCAmelCase   ,							__lowerCAmelCase     ):
				raise Exception('''Years to repay must be an integer > 0'''     )

		# Yearly rate is divided by 12 to get monthly rate
		snake_case__						:       int       						=  rate_per_annum / 12

		# Years to repay is multiplied by 12 to get number of payments as payment is monthly
		snake_case__						:       Tuple       						=  years_to_repay * 12

		return (
		    principal
		    * rate_per_month
		    * (1 + rate_per_month) ** number_of_payments
		    / ((1 + rate_per_month) ** number_of_payments - 1)
		)


if __name__ == "__main__":
					import doctest

					doctest.testmod()



