




from math import sqrt

def lowercase    (  SCREAMING_SNAKE_CASE )			->							bool:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (
			    number >= 0
			), "'number' must been an int and positive"

			SCREAMING_SNAKE_CASE_								=						True

			# 0 and 1 are none primes.
			if number <= 1:
						SCREAMING_SNAKE_CASE_								=						False

			for divisor in range(2       ,    int(round(sqrt(SCREAMING_SNAKE_CASE ) ) ) + 1 ):
						# if 'number' divisible by 'divisor' then sets 'status'
						# of false and break up the loop.
						if number % divisor == 0:
									SCREAMING_SNAKE_CASE_								=						False
									break

    # precondition
			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ), "'status' must been from type bool"

			return status

def lowercase    (  SCREAMING_SNAKE_CASE )			->							Any:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (n > 2), "'N' must been an int and > 2"

			# beginList: contains all natural numbers from 2 up to N
			SCREAMING_SNAKE_CASE_								=						list(range(2       ,    n + 1 ) )

			SCREAMING_SNAKE_CASE_								=						[]  # this list will be returns.

			# actual sieve of erathostenes
			for i in range(len(SCREAMING_SNAKE_CASE ) ):
						for j in range(i + 1       ,    len(SCREAMING_SNAKE_CASE ) ):
									if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
												SCREAMING_SNAKE_CASE_								=						0

    # filters actual prime numbers.
			SCREAMING_SNAKE_CASE_								=						[x for x in begin_list if x != 0]

			# precondition
			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ), "'ans' must been from type list"

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE )			->							Optional[Any]:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (n > 2), "'N' must been an int and > 2"

			SCREAMING_SNAKE_CASE_								=						[]

			# iterates over all numbers between 2 up to N+1
			# if a number is prime then appends to list 'ans'
			for number in range(2       ,    n + 1 ):
						if is_prime(SCREAMING_SNAKE_CASE ):
									ans.append(SCREAMING_SNAKE_CASE )

    # precondition
			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ), "'ans' must been from type list"

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE )			->							List[str]:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and number >= 0, "'number' must been an int and >= 0"

			SCREAMING_SNAKE_CASE_								=						[]  # this list will be returns of the function.

			# potential prime number factors.

			SCREAMING_SNAKE_CASE_								=						2

			SCREAMING_SNAKE_CASE_								=						number

			if number == 0 or number == 1:
						ans.append(SCREAMING_SNAKE_CASE )

			# if 'number' not prime then builds the prime factorization of 'number'
			elif not is_prime(SCREAMING_SNAKE_CASE ):
						while quotient != 1:
									if is_prime(SCREAMING_SNAKE_CASE ) and (quotient % factor == 0):
												ans.append(SCREAMING_SNAKE_CASE )
												quotient /= factor
									else:
												factor += 1

			else:
						ans.append(SCREAMING_SNAKE_CASE )

			# precondition
			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ), "'ans' must been from type list"

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE )			->							List[Any]:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (
			    number >= 0
			), "'number' bust been an int and >= 0"

			SCREAMING_SNAKE_CASE_								=						0

			# prime factorization of 'number'
			SCREAMING_SNAKE_CASE_								=						prime_factorization(SCREAMING_SNAKE_CASE )

			SCREAMING_SNAKE_CASE_								=						max(SCREAMING_SNAKE_CASE )

			# precondition
			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ), "'ans' must been from type int"

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE )			->							Optional[int]:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (
			    number >= 0
			), "'number' bust been an int and >= 0"

			SCREAMING_SNAKE_CASE_								=						0

			# prime factorization of 'number'
			SCREAMING_SNAKE_CASE_								=						prime_factorization(SCREAMING_SNAKE_CASE )

			SCREAMING_SNAKE_CASE_								=						min(SCREAMING_SNAKE_CASE )

			# precondition
			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ), "'ans' must been from type int"

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE )			->							List[str]:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ), "'number' must been an int"
			assert isinstance(number % 2 == 0       ,    SCREAMING_SNAKE_CASE ), "compare bust been from type bool"

			return number % 2 == 0

def lowercase    (  SCREAMING_SNAKE_CASE )			->							Optional[Any]:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ), "'number' must been an int"
			assert isinstance(number % 2 != 0       ,    SCREAMING_SNAKE_CASE ), "compare bust been from type bool"

			return number % 2 != 0

def lowercase    (  SCREAMING_SNAKE_CASE )			->							str:


			'''simple docstring'''







			assert (
			    isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (number > 2) and is_even(SCREAMING_SNAKE_CASE )
			), "'number' must been an int, even and > 2"

			SCREAMING_SNAKE_CASE_								=						[]  # this list will returned

			# creates a list of prime numbers between 2 up to 'number'
			SCREAMING_SNAKE_CASE_								=						get_prime_numbers(SCREAMING_SNAKE_CASE )
			SCREAMING_SNAKE_CASE_								=						len(SCREAMING_SNAKE_CASE )

			# run variable for while-loops.
			SCREAMING_SNAKE_CASE_								=						0
			SCREAMING_SNAKE_CASE_								=						None

			# exit variable. for break up the loops
			SCREAMING_SNAKE_CASE_								=						True

			while i < len_pn and loop:
						SCREAMING_SNAKE_CASE_								=						i + 1

						while j < len_pn and loop:
									if prime_numbers[i] + prime_numbers[j] == number:
												SCREAMING_SNAKE_CASE_								=						False
												ans.append(prime_numbers[i] )
												ans.append(prime_numbers[j] )

									j += 1

						i += 1

			# precondition
			assert (
			    isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and (len(SCREAMING_SNAKE_CASE ) == 2)
			    and (ans[0] + ans[1] == number)
			    and is_prime(ans[0] )
			    and is_prime(ans[1] )
			), "'ans' must contains two primes. And sum of elements must been eq 'number'"

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )			->							Any:


			'''simple docstring'''







			assert (
			    isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and (numbera >= 0)
			    and (numbera >= 0)
			), "'number1' and 'number2' must been positive integer."

			SCREAMING_SNAKE_CASE_								=						0

			while numbera != 0:
						SCREAMING_SNAKE_CASE_								=						numbera % numbera
						SCREAMING_SNAKE_CASE_								=						numbera
						SCREAMING_SNAKE_CASE_								=						rest

			# precondition
			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (
			    numbera >= 0
			), "'number' must been from type int and positive"

			return numbera

def lowercase    (  SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )			->							Tuple:


			'''simple docstring'''







			assert (
			    isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and (numbera >= 1)
			    and (numbera >= 1)
			), "'number1' and 'number2' must been positive integer."

			SCREAMING_SNAKE_CASE_								=						1  # actual answer that will be return.

			# for kgV (x,1)
			if numbera > 1 and numbera > 1:
						# builds the prime factorization of 'number1' and 'number2'
						SCREAMING_SNAKE_CASE_								=						prime_factorization(SCREAMING_SNAKE_CASE )
						SCREAMING_SNAKE_CASE_								=						prime_factorization(SCREAMING_SNAKE_CASE )

			elif numbera == 1 or numbera == 1:
						SCREAMING_SNAKE_CASE_								=						[]
						SCREAMING_SNAKE_CASE_								=						[]
						SCREAMING_SNAKE_CASE_								=						max(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )

			SCREAMING_SNAKE_CASE_								=						0
			SCREAMING_SNAKE_CASE_								=						0

			SCREAMING_SNAKE_CASE_								=						[]  # captured numbers int both 'primeFac1' and 'primeFac2'

			# iterates through primeFac1
			for n in prime_fac_a:
						if n not in done:
									if n in prime_fac_a:
												SCREAMING_SNAKE_CASE_								=						prime_fac_a.count(SCREAMING_SNAKE_CASE )
												SCREAMING_SNAKE_CASE_								=						prime_fac_a.count(SCREAMING_SNAKE_CASE )

												for _ in range(max(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) ):
															ans *= n

									else:
												SCREAMING_SNAKE_CASE_								=						prime_fac_a.count(SCREAMING_SNAKE_CASE )

												for _ in range(SCREAMING_SNAKE_CASE ):
															ans *= n

									done.append(SCREAMING_SNAKE_CASE )

    # iterates through primeFac2
			for n in prime_fac_a:
						if n not in done:
									SCREAMING_SNAKE_CASE_								=						prime_fac_a.count(SCREAMING_SNAKE_CASE )

									for _ in range(SCREAMING_SNAKE_CASE ):
												ans *= n

									done.append(SCREAMING_SNAKE_CASE )

    # precondition
			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (
			    ans >= 0
			), "'ans' must been from type int and positive"

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE )			->							str:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (n >= 0), "'number' must been a positive int"

			SCREAMING_SNAKE_CASE_								=						0
			SCREAMING_SNAKE_CASE_								=						2  # this variable holds the answer

			while index < n:
						index += 1

						ans += 1  # counts to the next number

						# if ans not prime then
						# runs to the next prime number.
						while not is_prime(SCREAMING_SNAKE_CASE ):
									ans += 1

    # precondition
			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and is_prime(
			    SCREAMING_SNAKE_CASE ), "'ans' must been a prime number and from type int"

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )			->							int:


			'''simple docstring'''







			assert (
			    is_prime(SCREAMING_SNAKE_CASE ) and is_prime(SCREAMING_SNAKE_CASE ) and (p_number_a < p_number_a)
			), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

			SCREAMING_SNAKE_CASE_								=						p_number_a + 1  # jump to the next number

			SCREAMING_SNAKE_CASE_								=						[]  # this list will be returns.

			# if number is not prime then
			# fetch the next prime number.
			while not is_prime(SCREAMING_SNAKE_CASE ):
						number += 1

			while number < p_number_a:
						ans.append(SCREAMING_SNAKE_CASE )

						number += 1

						# fetch the next prime number.
						while not is_prime(SCREAMING_SNAKE_CASE ):
									number += 1

    # precondition
			assert (
			    isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and ans[0] != p_number_a
			    and ans[len(SCREAMING_SNAKE_CASE ) - 1] != p_number_a
			), "'ans' must been a list without the arguments"

			# 'ans' contains not 'pNumber1' and 'pNumber2' !
			return ans

def lowercase    (  SCREAMING_SNAKE_CASE )			->							Optional[int]:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (n >= 1), "'n' must been int and >= 1"

			SCREAMING_SNAKE_CASE_								=						[]  # will be returned.

			for divisor in range(1       ,    n + 1 ):
						if n % divisor == 0:
									ans.append(SCREAMING_SNAKE_CASE )

    # precondition
			assert ans[0] == 1 and ans[len(SCREAMING_SNAKE_CASE ) - 1] == n, "Error in function getDivisiors(...)"

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE )			->							Union[str, Any]:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (
			    number > 1
			), "'number' must been an int and >= 1"

			SCREAMING_SNAKE_CASE_								=						get_divisors(SCREAMING_SNAKE_CASE )

			# precondition
			assert (
			    isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and (divisors[0] == 1)
			    and (divisors[len(SCREAMING_SNAKE_CASE ) - 1] == number)
			), "Error in help-function getDivisiors(...)"

			# summed all divisors up to 'number' (exclusive), hence [:-1]
			return sum(divisors[:-1] ) == number

def lowercase    (  SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )			->							Any:


			'''simple docstring'''







			assert (
			    isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and (denominator != 0)
			), "The arguments must been from type int and 'denominator' != 0"

			# build the greatest common divisor of numerator and denominator.
			SCREAMING_SNAKE_CASE_								=						gcd(abs(SCREAMING_SNAKE_CASE )       ,    abs(SCREAMING_SNAKE_CASE ) )

			# precondition
			assert (
			    isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )
			    and (numerator % gcd_of_fraction == 0)
			    and (denominator % gcd_of_fraction == 0)
			), "Error in function gcd(...,...)"

			return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)

def lowercase    (  SCREAMING_SNAKE_CASE )			->							Any:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (n >= 0), "'n' must been a int and >= 0"

			SCREAMING_SNAKE_CASE_								=						1  # this will be return.

			for factor in range(1       ,    n + 1 ):
						ans *= factor

			return ans

def lowercase    (  SCREAMING_SNAKE_CASE )			->							Dict:


			'''simple docstring'''







			assert isinstance(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE ) and (n >= 0), "'n' must been an int and >= 0"

			SCREAMING_SNAKE_CASE_								=						0
			SCREAMING_SNAKE_CASE_								=						1
			SCREAMING_SNAKE_CASE_								=						1  # this will be return

			for _ in range(n - 1 ):
						SCREAMING_SNAKE_CASE_								=						ans
						ans += fiba
						SCREAMING_SNAKE_CASE_								=						tmp

			return ans






import qiskit

def lowercase    (  SCREAMING_SNAKE_CASE = 2 )			->							qiskit.result.counts.Counts:


			'''simple docstring'''







			SCREAMING_SNAKE_CASE_								=						qubits

			# Using Aer's simulator
			SCREAMING_SNAKE_CASE_								=						qiskit.Aer.get_backend('aer_simulator' )

			# Creating a Quantum Circuit acting on the q register
			SCREAMING_SNAKE_CASE_								=						qiskit.QuantumCircuit(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE )

			# Adding a H gate on qubit 0 (now q0 in superposition)
			circuit.h(0 )

			for i in range(1       ,    SCREAMING_SNAKE_CASE ):
						# Adding CX (CNOT) gate
						circuit.cx(i - 1       ,    SCREAMING_SNAKE_CASE )

			# Mapping the quantum measurement to the classical bits
			circuit.measure(list(range(SCREAMING_SNAKE_CASE ) )       ,    list(range(SCREAMING_SNAKE_CASE ) ) )

			# Now measuring any one qubit would affect other qubits to collapse
			# their super position and have same state as the measured one.

			# Executing the circuit on the simulator
			SCREAMING_SNAKE_CASE_								=						qiskit.execute(SCREAMING_SNAKE_CASE       ,    SCREAMING_SNAKE_CASE       ,    shots=10_00 )

			return job.result().get_counts(SCREAMING_SNAKE_CASE )


if __name__ == "__main__":
				print(f"""Total count for various states are: {quantum_entanglement(3)}""")
