def 	__lowerCamelCase							(				__a    : int		)      ->       str:
	if not isinstance(__a      ,							__a		):
		_lowercase   =(
		    "operation can not be conducted on a object of type "
		    f'''{type(__a		).__name__}'''
		)
		raise TypeError(__a		)
	_lowercase   =""
	for _ in range(0      ,							32		):
		bit_string += str(number % 2		)
		_lowercase   =number >> 1
	return bit_string

def 	__lowerCamelCase							(				__a    : int		)      ->       str:
	if number < 0:
		raise ValueError("the value of input must be positive"		)
	elif isinstance(__a      ,							__a		):
		raise TypeError("Input value must be a 'int' type"		)
	elif isinstance(__a      ,							__a		):
		raise TypeError("'<' not supported between instances of 'str' and 'int'"		)
	_lowercase   =0
	# iterator over [1 to 32],since we are dealing with 32 bit integer
	for _ in range(1      ,							33		):
		# left shift the bits by unity
		_lowercase   =result << 1
		# get the end bit
		_lowercase   =number % 2
		# right shift the bits by unity
		_lowercase   =number >> 1
		# add that bit to our ans
		_lowercase   =result | end_bit
	return get_reverse_bit_string(__a		)


if __name__ == "__main__":
						import doctest

						doctest.testmod()







from __future__ import annotations






def        lowerCamelCase_  (				UpperCAmelCase_				:  int						)    ->					list[int]:




	'''simple docstring'''




	_UpperCamelCase :		Any								=    [0] * max_one

	for i in range(0				,							UpperCAmelCase_						):
		_UpperCamelCase :		List[Any]								=    i

	for i in range(2				,							UpperCAmelCase_						):
		if totients[i] == i:
			for j in range(UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_						):
				totients[j] -= totients[j] // i

	return totients






def        lowerCamelCase_  (				UpperCAmelCase_				:  int				,							UpperCAmelCase_				:  int						)    ->					bool:




	'''simple docstring'''




	return sorted(str(UpperCAmelCase_						)						) == sorted(str(UpperCAmelCase_						)						)






def        lowerCamelCase_  (				UpperCAmelCase_				:  int = 1_0_0_0_0_0_0_0						)    ->					int:




	'''simple docstring'''




	_UpperCamelCase :		List[Any]								=    1  # i
	_UpperCamelCase :		List[str]								=    0  # φ(i)
	_UpperCamelCase :		str								=    get_totients(max_n + 1						)

	for i in range(2				,							max_n + 1						):
		_UpperCamelCase :		List[str]								=    totients[i]

		if i * min_denominator < min_numerator * t and has_same_digits(UpperCAmelCase_				,							UpperCAmelCase_						):
			_UpperCamelCase :		Optional[int]								=    i
			_UpperCamelCase :		Tuple								=    t

	return min_numerator


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




