





import math






def 					lowerCamelCase_						(    _UpperCamelCase			)		->						bool:






				"""simple docstring"""


				if 1 < number < 4:
								# 2 and 3 are primes
								return True
				elif number < 2 or number % 2 == 0 or number % 3 == 0:
								# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
								return False

				# All primes number are in format of 6k +/- 1
				for i in range(5			,    int(math.sqrt(_UpperCamelCase			) + 1			)			,    6			):
								if number % i == 0 or number % (i + 2) == 0:
												return False
				return True






def 					lowerCamelCase_						(    _UpperCamelCase = 10_001			)		->						int:






				"""simple docstring"""


				try:
								snake_case_						:   str           =  int(_UpperCamelCase			)
				except (TypeError, ValueError):
								raise TypeError('''Parameter nth must be int or castable to int.'''			) from None
				if nth <= 0:
								raise ValueError('''Parameter nth must be greater than or equal to one.'''			)
				snake_case_						:   list[int]           =  []
				snake_case_						:   Union[str, Any]           =  2
				while len(_UpperCamelCase			) < nth:
								if is_prime(_UpperCamelCase			):
												primes.append(_UpperCamelCase			)
												num += 1
								else:
												num += 1
				return primes[len(_UpperCamelCase			) - 1]


if __name__ == "__main__":
	print(F'''{solution() = }''')







def 					lowerCamelCase_						(    _UpperCamelCase			)		->						int:






				"""simple docstring"""


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

				snake_case_						:   List[Any]           =  0
				snake_case_						:   Tuple           =  str(_UpperCamelCase			)

				while len(_UpperCamelCase			) != 1:
								snake_case_						:   Tuple           =  [int(_UpperCamelCase			) for i in num_string]

								snake_case_						:   Dict           =  1
								for i in range(0			,    len(_UpperCamelCase			)			):
												total *= numbers[i]

								snake_case_						:   str           =  str(_UpperCamelCase			)

								steps += 1
				return steps






def 					lowerCamelCase_						(    _UpperCamelCase			)		->						int:






				"""simple docstring"""


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

				snake_case_						:   Any           =  0
				snake_case_						:   Tuple           =  str(_UpperCamelCase			)

				while len(_UpperCamelCase			) != 1:
								snake_case_						:   List[str]           =  [int(_UpperCamelCase			) for i in num_string]

								snake_case_						:   Optional[int]           =  0
								for i in range(0			,    len(_UpperCamelCase			)			):
												total += numbers[i]

								snake_case_						:   Tuple           =  str(_UpperCamelCase			)

								steps += 1
				return steps


if __name__ == "__main__":
	import doctest

	doctest.testmod()
