



'''simple docstring'''

import qiskit
from qiskit.providers import Backend



def   __lowerCamelCase  (   lowerCAmelCase_		,    lowerCAmelCase_						)    ->   tuple[qiskit.QuantumCircuit, str, str]:
		_a  :   List[str]				 =							bin(lowerCAmelCase_						)[2:], bin(lowerCAmelCase_						)[2:]  # Remove leading '0b'

		# Ensure that both strings are of the same length
		if len(lowerCAmelCase_						) > len(lowerCAmelCase_						):
				_a  :   Union[str, Any]				 =							y.zfill(len(lowerCAmelCase_						)						)
		else:
				_a  :   str				 =							x.zfill(len(lowerCAmelCase_						)						)

		# We need (3 * number of bits in the larger number)+1 qBits
		# The second parameter is the number of classical registers, to measure the result
		_a  :   List[str]				 =							qiskit.QuantumCircuit((len(lowerCAmelCase_						) * 3) + 1		,    len(lowerCAmelCase_						) + 1						)

		# We are essentially "not-ing" the bits that are 1
		# Reversed because it's easier to perform ops on more significant bits
		for i in range(len(lowerCAmelCase_						)						):
				if x[::-1][i] == "1":
						circuit.x(lowerCAmelCase_						)
		for j in range(len(lowerCAmelCase_						)						):
				if y[::-1][j] == "1":
						circuit.x(len(lowerCAmelCase_						) + j						)

		return circuit, x, y



def   __lowerCamelCase  (   lowerCAmelCase_		,    lowerCAmelCase_		,    lowerCAmelCase_		,    lowerCAmelCase_		,    lowerCAmelCase_		,    )    ->   Optional[int]:
		circuit.ccx(lowerCAmelCase_		,    lowerCAmelCase_		,    lowerCAmelCase_						)
		circuit.cx(lowerCAmelCase_		,    lowerCAmelCase_						)
		circuit.ccx(lowerCAmelCase_		,    lowerCAmelCase_		,    lowerCAmelCase_						)
		circuit.cx(lowerCAmelCase_		,    lowerCAmelCase_						)
		circuit.cx(lowerCAmelCase_		,    lowerCAmelCase_						)



def   __lowerCamelCase  (   lowerCAmelCase_		,    lowerCAmelCase_		,    lowerCAmelCase_ = qiskit.Aer.get_backend('aer_simulator'						)		,    )    ->   int:

		if vala < 0 or vala < 0:
				raise ValueError('Both Integers must be positive!'						)

		# Store the Integers
		_a  :   Optional[Any]				 =							store_two_classics(lowerCAmelCase_		,    lowerCAmelCase_						)
		for i in range(len(lowerCAmelCase_						)						):
				full_adder(lowerCAmelCase_		,    lowerCAmelCase_		,    len(lowerCAmelCase_						) + i		,    len(lowerCAmelCase_						) + len(lowerCAmelCase_						) + i		,    len(lowerCAmelCase_						) + len(lowerCAmelCase_						) + i + 1						)
				circuit.barrier()  # Optional, just for aesthetics

		# Measure the resultant qBits
		for i in range(len(lowerCAmelCase_						) + 1						):
				circuit.measure([(len(lowerCAmelCase_						) * 2) + i]		,    [i]						)

		_a  :   Dict				 =							qiskit.execute(lowerCAmelCase_		,    lowerCAmelCase_		,    shots=1						).result()

		# The result is in binary. Convert it back to int
		return int(list(res.get_counts()						)[0]		,    2						)


if __name__ == "__main__":
							import doctest

							doctest.testmod()









'''simple docstring'''

import math



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



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



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



def   __lowerCamelCase  (   lowerCAmelCase_						)    ->   int:
		_a  :   Dict				 =							n
		for x in set(prime_factors(lowerCAmelCase_						)						):
				s *= (x - 1) / x
		return int(lowerCAmelCase_						)


if __name__ == "__main__":
							import doctest

							doctest.testmod()




