






def 					lowercase__				(					__snake_case   :							int  ):


					'''simple docstring'''


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

					UpperCAmelCase_   :				Union[str, Any]						=						0
					UpperCAmelCase_   :				Tuple						=						str(__snake_case  )

					while len(__snake_case  ) != 1:
										UpperCAmelCase_   :				Any						=						[int(__snake_case  ) for i in num_string]

										UpperCAmelCase_   :				Optional[int]						=						1
										for i in range(0					,	len(__snake_case  )  ):
															total *= numbers[i]

										UpperCAmelCase_   :				Tuple						=						str(__snake_case  )

										steps += 1
					return steps




def 					lowercase__				(					__snake_case   :							int  ):


					'''simple docstring'''


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

					UpperCAmelCase_   :				Tuple						=						0
					UpperCAmelCase_   :				str						=						str(__snake_case  )

					while len(__snake_case  ) != 1:
										UpperCAmelCase_   :				Optional[int]						=						[int(__snake_case  ) for i in num_string]

										UpperCAmelCase_   :				Tuple						=						0
										for i in range(0					,	len(__snake_case  )  ):
															total += numbers[i]

										UpperCAmelCase_   :				Any						=						str(__snake_case  )

										steps += 1
					return steps


if __name__ == "__main__":
		import doctest

		doctest.testmod()











def 					lowercase__				(					__snake_case   :							int  ):


					'''simple docstring'''


					UpperCAmelCase_   :				Tuple						=						[1]

					UpperCAmelCase_	,  UpperCAmelCase_	,  UpperCAmelCase_   :				Tuple						=						0, 0, 0
					UpperCAmelCase_   :				Union[str, Any]						=						ugly_nums[ia] * 2
					UpperCAmelCase_   :				Tuple						=						ugly_nums[ia] * 3
					UpperCAmelCase_   :				Union[str, Any]						=						ugly_nums[ia] * 5

					for _ in range(1					,	__snake_case  ):
										UpperCAmelCase_   :				Tuple						=						min(__snake_case					,	__snake_case					,	__snake_case  )
										ugly_nums.append(__snake_case  )
										if next_num == next_a:
															ia += 1
															UpperCAmelCase_   :				Union[str, Any]						=						ugly_nums[ia] * 2
										if next_num == next_a:
															ia += 1
															UpperCAmelCase_   :				Any						=						ugly_nums[ia] * 3
										if next_num == next_a:
															ia += 1
															UpperCAmelCase_   :				List[str]						=						ugly_nums[ia] * 5
					return ugly_nums[-1]


if __name__ == "__main__":
		from doctest import testmod

		testmod(verbose=True)
		print(F'{ugly_numbers(200) = }')






