
"""simple docstring"""




from math import sqrt



def        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		assert isinstance(__UpperCamelCase,							__UpperCamelCase    ) 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(__UpperCamelCase    )    )    ) + 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(__UpperCamelCase,							__UpperCamelCase    ), "'status' must been from type bool"

		return status



def        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		assert isinstance(__UpperCamelCase,							__UpperCamelCase    ) 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(__UpperCamelCase    )    ):
				for j in range(i + 1,							len(__UpperCamelCase    )    ):
						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(__UpperCamelCase,							__UpperCamelCase    ), "'ans' must been from type list"

		return ans



def        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		assert isinstance(__UpperCamelCase,							__UpperCamelCase    ) 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(__UpperCamelCase    ):
						ans.append(__UpperCamelCase    )

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

		return ans



def        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		assert isinstance(__UpperCamelCase,							__UpperCamelCase    ) 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(__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        _lowerCamelCase ( __a    ):

		'''simple docstring'''





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

		SCREAMING_SNAKE_CASE_						 = 0

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

		SCREAMING_SNAKE_CASE_						 = max(__UpperCamelCase    )

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

		return ans



def        _lowerCamelCase ( __a    ):

		'''simple docstring'''





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

		SCREAMING_SNAKE_CASE_						 = 0

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

		SCREAMING_SNAKE_CASE_						 = min(__UpperCamelCase    )

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

		return ans



def        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		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        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		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        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		assert (
		    isinstance(__UpperCamelCase,							__UpperCamelCase    ) and (number > 2) and is_even(__UpperCamelCase    )
		), "'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(__UpperCamelCase    )
		SCREAMING_SNAKE_CASE_						 = len(__UpperCamelCase    )

		# 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(__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        _lowerCamelCase ( __a,							__a    ):

		'''simple docstring'''





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

		return numbera



def        _lowerCamelCase ( __a,							__a    ):

		'''simple docstring'''





		assert (
		    isinstance(__UpperCamelCase,							__UpperCamelCase    )
		    and isinstance(__UpperCamelCase,							__UpperCamelCase    )
		    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(__UpperCamelCase    )
				SCREAMING_SNAKE_CASE_						 = prime_factorization(__UpperCamelCase    )

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

		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(__UpperCamelCase    )
								SCREAMING_SNAKE_CASE_						 = prime_fac_a.count(__UpperCamelCase    )

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

						else:
								SCREAMING_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:
						SCREAMING_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        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		assert isinstance(__UpperCamelCase,							__UpperCamelCase    ) 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(__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        _lowerCamelCase ( __a,							__a    ):

		'''simple docstring'''





		assert (
		    is_prime(__UpperCamelCase    ) and is_prime(__UpperCamelCase    ) 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(__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        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		assert isinstance(__UpperCamelCase,							__UpperCamelCase    ) 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(__UpperCamelCase    )

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

		return ans



def        _lowerCamelCase ( __a    ):

		'''simple docstring'''





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

		SCREAMING_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        _lowerCamelCase ( __a,							__a    ):

		'''simple docstring'''





		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.
		SCREAMING_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        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		assert isinstance(__UpperCamelCase,							__UpperCamelCase    ) 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        _lowerCamelCase ( __a    ):

		'''simple docstring'''





		assert isinstance(__UpperCamelCase,							__UpperCamelCase    ) 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

"""simple docstring"""




def        _lowerCamelCase ( __a    ):

		if not isinstance(__a,							__a    ):
				SCREAMING_SNAKE_CASE_						 = F'Input value of [number={number}] must be an integer'
				raise TypeError(__a    )

		if number < 1:
				SCREAMING_SNAKE_CASE_						 = F'Input value of [number={number}] must be > 0'
				raise ValueError(__a    )

		SCREAMING_SNAKE_CASE_						 = 1

		for i in range(1,							__a    ):
				current_number *= 4 * i - 2
				current_number //= i + 1

		return current_number


if __name__ == "__main__":
						import doctest

						doctest.testmod()