

"""simple docstring"""


from math import sqrt






def    a__   (   __SCREAMING_SNAKE_CASE			)     ->     bool:
		assert isinstance(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			) and (
		    number >= 0
		), "'number' must been an int and positive"

		__lowerCAmelCase:	Dict   				=      True

		# 0 and 1 are none primes.
		if number <= 1:
				__lowerCAmelCase:	Optional[Any]   				=      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:
						__lowerCAmelCase:	List[Any]   				=      False
						break

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

		return status






def    a__   (   __SCREAMING_SNAKE_CASE			)     ->     Any:
		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
		__lowerCAmelCase:	Union[str, Any]   				=      list(range(2 ,			n + 1			)			)

		__lowerCAmelCase:	Any   				=      []  # 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):
								__lowerCAmelCase:	List[Any]   				=      0

    # filters actual prime numbers.
		__lowerCAmelCase:	Optional[int]   				=      [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    a__   (   __SCREAMING_SNAKE_CASE			)     ->     Any:
		assert isinstance(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			) and (n > 2), "'N' must been an int and > 2"

		__lowerCAmelCase:	Union[str, Any]   				=      []

		# 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    a__   (   __SCREAMING_SNAKE_CASE			)     ->     int:
		assert isinstance(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			) and number >= 0, "'number' must been an int and >= 0"

		__lowerCAmelCase:	Tuple   				=      []  # this list will be returns of the function.

		# potential prime number factors.

		__lowerCAmelCase:	str   				=      2

		__lowerCAmelCase:	Tuple   				=      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    a__   (   __SCREAMING_SNAKE_CASE			)     ->     Any:
		assert isinstance(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			) and (
		    number >= 0
		), "'number' bust been an int and >= 0"

		__lowerCAmelCase:	Union[str, Any]   				=      0

		# prime factorization of 'number'
		__lowerCAmelCase:	str   				=      prime_factorization(__SCREAMING_SNAKE_CASE			)

		__lowerCAmelCase:	Dict   				=      max(__SCREAMING_SNAKE_CASE			)

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

		return ans






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

		__lowerCAmelCase:	List[str]   				=      0

		# prime factorization of 'number'
		__lowerCAmelCase:	Dict   				=      prime_factorization(__SCREAMING_SNAKE_CASE			)

		__lowerCAmelCase:	List[Any]   				=      min(__SCREAMING_SNAKE_CASE			)

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

		return ans






def    a__   (   __SCREAMING_SNAKE_CASE			)     ->     List[Any]:
		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    a__   (   __SCREAMING_SNAKE_CASE			)     ->     Any:
		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    a__   (   __SCREAMING_SNAKE_CASE			)     ->     List[Any]:
		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"

		__lowerCAmelCase:	Dict   				=      []  # this list will returned

		# creates a list of prime numbers between 2 up to 'number'
		__lowerCAmelCase:	Union[str, Any]   				=      get_prime_numbers(__SCREAMING_SNAKE_CASE			)
		__lowerCAmelCase:	Optional[Any]   				=      len(__SCREAMING_SNAKE_CASE			)

		# run variable for while-loops.
		__lowerCAmelCase:	List[str]   				=      0
		__lowerCAmelCase:	Any   				=      None

		# exit variable. for break up the loops
		__lowerCAmelCase:	Tuple   				=      True

		while i < len_pn and loop:
				__lowerCAmelCase:	str   				=      i + 1

				while j < len_pn and loop:
						if prime_numbers[i] + prime_numbers[j] == number:
								__lowerCAmelCase:	Union[str, Any]   				=      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    a__   (   __SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			)     ->     Union[str, Any]:
		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."

		__lowerCAmelCase:	Optional[int]   				=      0

		while numbera != 0:
				__lowerCAmelCase:	int   				=      numbera % numbera
				__lowerCAmelCase:	Tuple   				=      numbera
				__lowerCAmelCase:	Union[str, Any]   				=      rest

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

		return numbera






def    a__   (   __SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			)     ->     List[str]:
		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."

		__lowerCAmelCase:	int   				=      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'
				__lowerCAmelCase:	Optional[int]   				=      prime_factorization(__SCREAMING_SNAKE_CASE			)
				__lowerCAmelCase:	Union[str, Any]   				=      prime_factorization(__SCREAMING_SNAKE_CASE			)

		elif numbera == 1 or numbera == 1:
				__lowerCAmelCase:	Tuple   				=      []
				__lowerCAmelCase:	Optional[Any]   				=      []
				__lowerCAmelCase:	Union[str, Any]   				=      max(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			)

		__lowerCAmelCase:	List[str]   				=      0
		__lowerCAmelCase:	Optional[Any]   				=      0

		__lowerCAmelCase:	Dict   				=      []  # 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:
								__lowerCAmelCase:	int   				=      prime_fac_a.count(__SCREAMING_SNAKE_CASE			)
								__lowerCAmelCase:	int   				=      prime_fac_a.count(__SCREAMING_SNAKE_CASE			)

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

						else:
								__lowerCAmelCase:	Optional[Any]   				=      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:
						__lowerCAmelCase:	Tuple   				=      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    a__   (   __SCREAMING_SNAKE_CASE			)     ->     Any:
		assert isinstance(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			) and (n >= 0), "'number' must been a positive int"

		__lowerCAmelCase:	Union[str, Any]   				=      0
		__lowerCAmelCase:	Union[str, Any]   				=      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    a__   (   __SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			)     ->     Union[str, Any]:
		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'"

		__lowerCAmelCase:	Tuple   				=      p_number_a + 1  # jump to the next number

		__lowerCAmelCase:	int   				=      []  # 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    a__   (   __SCREAMING_SNAKE_CASE			)     ->     List[Any]:
		assert isinstance(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			) and (n >= 1), "'n' must been int and >= 1"

		__lowerCAmelCase:	Dict   				=      []  # 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    a__   (   __SCREAMING_SNAKE_CASE			)     ->     Union[str, Any]:
		assert isinstance(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			) and (
		    number > 1
		), "'number' must been an int and >= 1"

		__lowerCAmelCase:	List[str]   				=      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    a__   (   __SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			)     ->     List[Any]:
		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.
		__lowerCAmelCase:	Dict   				=      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    a__   (   __SCREAMING_SNAKE_CASE			)     ->     Tuple:
		assert isinstance(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			) and (n >= 0), "'n' must been a int and >= 0"

		__lowerCAmelCase:	Union[str, Any]   				=      1  # this will be return.

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

		return ans






def    a__   (   __SCREAMING_SNAKE_CASE			)     ->     Any:
		assert isinstance(__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			) and (n >= 0), "'n' must been an int and >= 0"

		__lowerCAmelCase:	int   				=      0
		__lowerCAmelCase:	int   				=      1
		__lowerCAmelCase:	List[Any]   				=      1  # this will be return

		for _ in range(n - 1			):
				__lowerCAmelCase:	List[str]   				=      ans
				ans += fiba
				__lowerCAmelCase:	Any   				=      tmp

		return ans





"""simple docstring"""


from __future__ import annotations

from math import pi






def    a__   (   __SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE			)     ->     dict[str, float]:

		if (inductance, frequency, reactance).count(0			) != 1:
				raise ValueError("One and only one argument must be 0"			)
		if inductance < 0:
				raise ValueError("Inductance cannot be negative"			)
		if frequency < 0:
				raise ValueError("Frequency cannot be negative"			)
		if reactance < 0:
				raise ValueError("Inductive reactance cannot be negative"			)
		if inductance == 0:
				return {"inductance": reactance / (2 * pi * frequency)}
		elif frequency == 0:
				return {"frequency": reactance / (2 * pi * inductance)}
		elif reactance == 0:
				return {"reactance": 2 * pi * frequency * inductance}
		else:
				raise ValueError("Exactly one argument must be 0"			)


if __name__ == "__main__":
			import doctest

			doctest.testmod()


