

def 							__SCREAMING_SNAKE_CASE							( lowerCAmelCase:     int     ,					lowerCAmelCase:     int = 1000       )      ->				str:
				if (not isinstance(lowerCAmelCase     ,					lowerCAmelCase       )) or (digit_position <= 0):
								raise ValueError("Digit position must be a positive integer"       )
				elif (not isinstance(lowerCAmelCase     ,					lowerCAmelCase       )) or (precision < 0):
								raise ValueError("Precision must be a nonnegative integer"       )

				# compute an approximation of (16 ** (n - 1)) * pi whose fractional part is mostly
				# accurate
				_UpperCAmelCase				:						Optional[Any]       						=			(
				    4 * _subsum(lowerCAmelCase     ,					1     ,					lowerCAmelCase       )
				    - 2 * _subsum(lowerCAmelCase     ,					4     ,					lowerCAmelCase       )
				    - _subsum(lowerCAmelCase     ,					5     ,					lowerCAmelCase       )
				    - _subsum(lowerCAmelCase     ,					6     ,					lowerCAmelCase       )
				)

				# return the first hex digit of the fractional part of the result
				return hex(int((sum_result % 1) * 16       )       )[2:]

def 							__SCREAMING_SNAKE_CASE							( lowerCAmelCase:     int     ,					lowerCAmelCase:     int     ,					lowerCAmelCase:     int       )      ->				float:
				_UpperCAmelCase				:						List[Any]       						=			0.0
				for sum_index in range(digit_pos_to_extract + precision       ):
								_UpperCAmelCase				:						Dict       						=			8 * sum_index + denominator_addend
								if sum_index < digit_pos_to_extract:
												# if the exponential term is an integer and we mod it by the denominator
												# before dividing, only the integer part of the sum will change;
												# the fractional part will not
												_UpperCAmelCase				:						List[Any]       						=			pow(
												    16     ,					digit_pos_to_extract - 1 - sum_index     ,					lowerCAmelCase       )
								else:
												_UpperCAmelCase				:						List[str]       						=			pow(16     ,					digit_pos_to_extract - 1 - sum_index       )
								total += exponential_term / denominator
				return total


if __name__ == "__main__":
					import doctest

					doctest.testmod()





import math

def 							__SCREAMING_SNAKE_CASE							( lowerCAmelCase:     int       )      ->				list:
				if n <= 0:
								raise ValueError("Only positive integers have prime factors"       )
				_UpperCAmelCase				:						Union[str, Any]       						=			[]
				while n % 2 == 0:
								pf.append(2       )
								_UpperCAmelCase				:						Tuple       						=			int(n / 2       )
				for i in range(3     ,					int(math.sqrt(lowerCAmelCase       )       ) + 1     ,					2       ):
								while n % i == 0:
												pf.append(lowerCAmelCase       )
												_UpperCAmelCase				:						Optional[int]       						=			int(n / i       )
				if n > 2:
								pf.append(lowerCAmelCase       )
				return pf

def 							__SCREAMING_SNAKE_CASE							( lowerCAmelCase:     int       )      ->				int:
				if n <= 0:
								raise ValueError("Only positive numbers are accepted"       )
				_UpperCAmelCase				:						Dict       						=			1
				_UpperCAmelCase				:						str       						=			1
				while n % 2 == 0:
								temp += 1
								_UpperCAmelCase				:						List[str]       						=			int(n / 2       )
				div *= temp
				for i in range(3     ,					int(math.sqrt(lowerCAmelCase       )       ) + 1     ,					2       ):
								_UpperCAmelCase				:						Any       						=			1
								while n % i == 0:
												temp += 1
												_UpperCAmelCase				:						Any       						=			int(n / i       )
								div *= temp
				if n > 1:
								div *= 2
				return div

def 							__SCREAMING_SNAKE_CASE							( lowerCAmelCase:     int       )      ->				int:
				if n <= 0:
								raise ValueError("Only positive numbers are accepted"       )
				_UpperCAmelCase				:						Optional[Any]       						=			1
				_UpperCAmelCase				:						Union[str, Any]       						=			1
				while n % 2 == 0:
								temp += 1
								_UpperCAmelCase				:						Optional[Any]       						=			int(n / 2       )
				if temp > 1:
								s *= (2**temp - 1) / (2 - 1)
				for i in range(3     ,					int(math.sqrt(lowerCAmelCase       )       ) + 1     ,					2       ):
								_UpperCAmelCase				:						List[str]       						=			1
								while n % i == 0:
												temp += 1
												_UpperCAmelCase				:						List[Any]       						=			int(n / i       )
								if temp > 1:
												s *= (i**temp - 1) / (i - 1)
				return int(lowerCAmelCase       )

def 							__SCREAMING_SNAKE_CASE							( lowerCAmelCase:     int       )      ->				int:
				_UpperCAmelCase				:						Any       						=			n
				for x in set(prime_factors(lowerCAmelCase       )       ):
								s *= (x - 1) / x
				return int(lowerCAmelCase       )


if __name__ == "__main__":
					import doctest

					doctest.testmod()


