



from math import sqrt



def 	__a					(   lowerCAmelCase_       : int    )				->			bool:






						'''simple docstring'''



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

						UpperCAmelCase_=       True

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

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

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

						return status



def 	__a					(   lowerCAmelCase_       : List[Any]    )				->			Dict:






						'''simple docstring'''



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

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

						UpperCAmelCase_=       []  # this list will be returns.

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

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

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

						return ans



def 	__a					(   lowerCAmelCase_       : Tuple    )				->			Optional[int]:






						'''simple docstring'''



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

						UpperCAmelCase_=       []

						# 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(lowerCAmelCase_    ):
																		ans.append(lowerCAmelCase_    )

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

						return ans



def 	__a					(   lowerCAmelCase_       : int    )				->			Dict:






						'''simple docstring'''



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

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

						# potential prime number factors.

						UpperCAmelCase_=       2

						UpperCAmelCase_=       number

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

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

						else:
												ans.append(lowerCAmelCase_    )

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

						return ans



def 	__a					(   lowerCAmelCase_       : Optional[int]    )				->			List[Any]:






						'''simple docstring'''



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

						UpperCAmelCase_=       0

						# prime factorization of 'number'
						UpperCAmelCase_=       prime_factorization(lowerCAmelCase_    )

						UpperCAmelCase_=       max(lowerCAmelCase_    )

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

						return ans



def 	__a					(   lowerCAmelCase_       : Dict    )				->			List[str]:






						'''simple docstring'''



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

						UpperCAmelCase_=       0

						# prime factorization of 'number'
						UpperCAmelCase_=       prime_factorization(lowerCAmelCase_    )

						UpperCAmelCase_=       min(lowerCAmelCase_    )

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

						return ans



def 	__a					(   lowerCAmelCase_       : int    )				->			Optional[int]:






						'''simple docstring'''



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

						return number % 2 == 0



def 	__a					(   lowerCAmelCase_       : Optional[Any]    )				->			str:






						'''simple docstring'''



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

						return number % 2 != 0



def 	__a					(   lowerCAmelCase_       : List[str]    )				->			str:






						'''simple docstring'''



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

						UpperCAmelCase_=       []  # this list will returned

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

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

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

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

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

																		j += 1

												i += 1

						# precondition
						assert (
						    isinstance(lowerCAmelCase_      ,lowerCAmelCase_    )
						    and (len(lowerCAmelCase_    ) == 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 	__a					(   lowerCAmelCase_       : Union[str, Any]      ,lowerCAmelCase_       : Dict    )				->			Union[str, Any]:






						'''simple docstring'''



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

						UpperCAmelCase_=       0

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

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

						return numbera



def 	__a					(   lowerCAmelCase_       : int      ,lowerCAmelCase_       : Optional[int]    )				->			List[str]:






						'''simple docstring'''



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

						UpperCAmelCase_=       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'
												UpperCAmelCase_=       prime_factorization(lowerCAmelCase_    )
												UpperCAmelCase_=       prime_factorization(lowerCAmelCase_    )

						elif numbera == 1 or numbera == 1:
												UpperCAmelCase_=       []
												UpperCAmelCase_=       []
												UpperCAmelCase_=       max(lowerCAmelCase_      ,lowerCAmelCase_    )

						UpperCAmelCase_=       0
						UpperCAmelCase_=       0

						UpperCAmelCase_=       []  # 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:
																								UpperCAmelCase_=       prime_fac_a.count(lowerCAmelCase_    )
																								UpperCAmelCase_=       prime_fac_a.count(lowerCAmelCase_    )

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

																		else:
																								UpperCAmelCase_=       prime_fac_a.count(lowerCAmelCase_    )

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

																		done.append(lowerCAmelCase_    )

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

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

																		done.append(lowerCAmelCase_    )

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

						return ans



def 	__a					(   lowerCAmelCase_       : str    )				->			Union[str, Any]:






						'''simple docstring'''



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

						UpperCAmelCase_=       0
						UpperCAmelCase_=       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(lowerCAmelCase_    ):
																		ans += 1

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

						return ans



def 	__a					(   lowerCAmelCase_       : Any      ,lowerCAmelCase_       : List[Any]    )				->			Any:






						'''simple docstring'''



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

						UpperCAmelCase_=       p_number_a + 1  # jump to the next number

						UpperCAmelCase_=       []  # this list will be returns.

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

						while number < p_number_a:
												ans.append(lowerCAmelCase_    )

												number += 1

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

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

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



def 	__a					(   lowerCAmelCase_       : Tuple    )				->			Tuple:






						'''simple docstring'''



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

						UpperCAmelCase_=       []  # will be returned.

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

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

						return ans



def 	__a					(   lowerCAmelCase_       : Any    )				->			Tuple:






						'''simple docstring'''



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

						UpperCAmelCase_=       get_divisors(lowerCAmelCase_    )

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

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



def 	__a					(   lowerCAmelCase_       : List[str]      ,lowerCAmelCase_       : Optional[Any]    )				->			Optional[int]:






						'''simple docstring'''



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

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

						# precondition
						assert (
						    isinstance(lowerCAmelCase_      ,lowerCAmelCase_    )
						    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 	__a					(   lowerCAmelCase_       : Optional[int]    )				->			Optional[Any]:






						'''simple docstring'''



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

						UpperCAmelCase_=       1  # this will be return.

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

						return ans



def 	__a					(   lowerCAmelCase_       : Any    )				->			Optional[int]:






						'''simple docstring'''



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

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

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

						return ans









import warnings
from functools import wraps
from typing import Callable



def 	__a					(   lowerCAmelCase_       : Callable    )				->			Callable:






						'''simple docstring'''



						@wraps(lowerCAmelCase_    )
						def _inner_fn(*lowerCAmelCase_       : List[Any]      ,**lowerCAmelCase_       : Tuple    ):
												warnings.warn(
												    (F"""'{fn.__name__}' is experimental and might be subject to breaking changes in the future.""")      ,lowerCAmelCase_      ,)
												return fn(*lowerCAmelCase_      ,**lowerCAmelCase_    )

						return _inner_fn




