




'''simple docstring'''

from math import sqrt


def __UpperCAmelCase	(   _UpperCAmelCase		:   Dict						)      ->      Any:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (
			    number >= 0
			), "'number' must been an int and positive"

			__snake_case			=		True

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

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

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

			return status


def __UpperCAmelCase	(   _UpperCAmelCase		:   Union[str, Any]						)      ->      Union[str, Any]:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (n > 2), "'N' must been an int and > 2"

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

			__snake_case			=		[]  # this list will be returns.

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

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

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

			return ans


def __UpperCAmelCase	(   _UpperCAmelCase		:   Dict						)      ->      Optional[Any]:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (n > 2), "'N' must been an int and > 2"

			__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(_UpperCAmelCase						):
									ans.append(_UpperCAmelCase						)

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

			return ans


def __UpperCAmelCase	(   _UpperCAmelCase		:   Tuple						)      ->      Optional[int]:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and number >= 0, "'number' must been an int and >= 0"

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

			# potential prime number factors.

			__snake_case			=		2

			__snake_case			=		number

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

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

			else:
						ans.append(_UpperCAmelCase						)

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

			return ans


def __UpperCAmelCase	(   _UpperCAmelCase		:   Any						)      ->      Optional[Any]:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (
			    number >= 0
			), "'number' bust been an int and >= 0"

			__snake_case			=		0

			# prime factorization of 'number'
			__snake_case			=		prime_factorization(_UpperCAmelCase						)

			__snake_case			=		max(_UpperCAmelCase						)

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

			return ans


def __UpperCAmelCase	(   _UpperCAmelCase		:   Optional[int]						)      ->      Optional[int]:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (
			    number >= 0
			), "'number' bust been an int and >= 0"

			__snake_case			=		0

			# prime factorization of 'number'
			__snake_case			=		prime_factorization(_UpperCAmelCase						)

			__snake_case			=		min(_UpperCAmelCase						)

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

			return ans


def __UpperCAmelCase	(   _UpperCAmelCase		:   Any						)      ->      List[Any]:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						), "'number' must been an int"
			assert isinstance(number % 2 == 0		,				_UpperCAmelCase						), "compare bust been from type bool"

			return number % 2 == 0


def __UpperCAmelCase	(   _UpperCAmelCase		:   Tuple						)      ->      Tuple:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						), "'number' must been an int"
			assert isinstance(number % 2 != 0		,				_UpperCAmelCase						), "compare bust been from type bool"

			return number % 2 != 0


def __UpperCAmelCase	(   _UpperCAmelCase		:   Dict						)      ->      Dict:
			assert (
			    isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (number > 2) and is_even(_UpperCAmelCase						)
			), "'number' must been an int, even and > 2"

			__snake_case			=		[]  # this list will returned

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

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

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

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

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

									j += 1

						i += 1

			# precondition
			assert (
			    isinstance(_UpperCAmelCase		,				_UpperCAmelCase						)
			    and (len(_UpperCAmelCase						) == 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 __UpperCAmelCase	(   _UpperCAmelCase		:   List[Any]		,				_UpperCAmelCase		:   Any						)      ->      List[str]:
			assert (
			    isinstance(_UpperCAmelCase		,				_UpperCAmelCase						)
			    and isinstance(_UpperCAmelCase		,				_UpperCAmelCase						)
			    and (numbera >= 0)
			    and (numbera >= 0)
			), "'number1' and 'number2' must been positive integer."

			__snake_case			=		0

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

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

			return numbera


def __UpperCAmelCase	(   _UpperCAmelCase		:   Optional[Any]		,				_UpperCAmelCase		:   List[Any]						)      ->      str:
			assert (
			    isinstance(_UpperCAmelCase		,				_UpperCAmelCase						)
			    and isinstance(_UpperCAmelCase		,				_UpperCAmelCase						)
			    and (numbera >= 1)
			    and (numbera >= 1)
			), "'number1' and 'number2' must been positive integer."

			__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'
						__snake_case			=		prime_factorization(_UpperCAmelCase						)
						__snake_case			=		prime_factorization(_UpperCAmelCase						)

			elif numbera == 1 or numbera == 1:
						__snake_case			=		[]
						__snake_case			=		[]
						__snake_case			=		max(_UpperCAmelCase		,				_UpperCAmelCase						)

			__snake_case			=		0
			__snake_case			=		0

			__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:
												__snake_case			=		prime_fac_a.count(_UpperCAmelCase						)
												__snake_case			=		prime_fac_a.count(_UpperCAmelCase						)

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

									else:
												__snake_case			=		prime_fac_a.count(_UpperCAmelCase						)

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

									done.append(_UpperCAmelCase						)

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

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

									done.append(_UpperCAmelCase						)

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

			return ans


def __UpperCAmelCase	(   _UpperCAmelCase		:   Optional[Any]						)      ->      str:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (n >= 0), "'number' must been a positive int"

			__snake_case			=		0
			__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(_UpperCAmelCase						):
									ans += 1

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

			return ans


def __UpperCAmelCase	(   _UpperCAmelCase		:   Optional[Any]		,				_UpperCAmelCase		:   Optional[Any]						)      ->      List[str]:
			assert (
			    is_prime(_UpperCAmelCase						) and is_prime(_UpperCAmelCase						) and (p_number_a < p_number_a)
			), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

			__snake_case			=		p_number_a + 1  # jump to the next number

			__snake_case			=		[]  # this list will be returns.

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

			while number < p_number_a:
						ans.append(_UpperCAmelCase						)

						number += 1

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

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

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


def __UpperCAmelCase	(   _UpperCAmelCase		:   Dict						)      ->      Union[str, Any]:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (n >= 1), "'n' must been int and >= 1"

			__snake_case			=		[]  # will be returned.

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

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

			return ans


def __UpperCAmelCase	(   _UpperCAmelCase		:   Tuple						)      ->      str:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (
			    number > 1
			), "'number' must been an int and >= 1"

			__snake_case			=		get_divisors(_UpperCAmelCase						)

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

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


def __UpperCAmelCase	(   _UpperCAmelCase		:   List[str]		,				_UpperCAmelCase		:   int						)      ->      int:
			assert (
			    isinstance(_UpperCAmelCase		,				_UpperCAmelCase						)
			    and isinstance(_UpperCAmelCase		,				_UpperCAmelCase						)
			    and (denominator != 0)
			), "The arguments must been from type int and 'denominator' != 0"

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

			# precondition
			assert (
			    isinstance(_UpperCAmelCase		,				_UpperCAmelCase						)
			    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 __UpperCAmelCase	(   _UpperCAmelCase		:   int						)      ->      int:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (n >= 0), "'n' must been a int and >= 0"

			__snake_case			=		1  # this will be return.

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

			return ans






def __UpperCAmelCase	(   _UpperCAmelCase		:   List[Any]						)      ->      List[Any]:
			assert isinstance(_UpperCAmelCase		,				_UpperCAmelCase						) and (n >= 0), "'n' must been an int and >= 0"

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

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

			return ans











'''simple docstring'''


def 						lowerCamelCase__   (				a							):
						if number < 0:
												raise ValueError('number must not be negative'							)
						return number & (number - 1) == 0


if __name__ == "__main__":
	import doctest

	doctest.testmod()

