





from math import sqrt




def    _lowerCAmelCase						(				lowerCamelCase_			:  int				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (
			    number >= 0
			), "'number' must been an int and positive"

			__lowercase										=							True

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

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

			return status




def    _lowerCAmelCase						(				lowerCamelCase_			:  Optional[Any]				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (n > 2), "'N' must been an int and > 2"

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

			__lowercase										=							[]  # 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):
												__lowercase										=							0

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

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

			return ans




def    _lowerCAmelCase						(				lowerCamelCase_			:  Any				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (n > 2), "'N' must been an int and > 2"

			__lowercase										=							[]

			# 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    _lowerCAmelCase						(				lowerCamelCase_			:  List[str]				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and number >= 0, "'number' must been an int and >= 0"

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

			# potential prime number factors.

			__lowercase										=							2

			__lowercase										=							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    _lowerCAmelCase						(				lowerCamelCase_			:  Any				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (
			    number >= 0
			), "'number' bust been an int and >= 0"

			__lowercase										=							0

			# prime factorization of 'number'
			__lowercase										=							prime_factorization(lowerCamelCase_				)

			__lowercase										=							max(lowerCamelCase_				)

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

			return ans




def    _lowerCAmelCase						(				lowerCamelCase_			:  List[Any]				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (
			    number >= 0
			), "'number' bust been an int and >= 0"

			__lowercase										=							0

			# prime factorization of 'number'
			__lowercase										=							prime_factorization(lowerCamelCase_				)

			__lowercase										=							min(lowerCamelCase_				)

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

			return ans




def    _lowerCAmelCase						(				lowerCamelCase_			:  Tuple				):
			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    _lowerCAmelCase						(				lowerCamelCase_			:  Optional[Any]				):
			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    _lowerCAmelCase						(				lowerCamelCase_			:  int				):
			assert (
			    isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (number > 2) and is_even(lowerCamelCase_				)
			), "'number' must been an int, even and > 2"

			__lowercase										=							[]  # this list will returned

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

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

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

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

						while j < len_pn and loop:
									if prime_numbers[i] + prime_numbers[j] == number:
												__lowercase										=							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    _lowerCAmelCase						(				lowerCamelCase_			:  Union[str, Any]		,       lowerCamelCase_			:  Union[str, Any]				):
			assert (
			    isinstance(lowerCamelCase_		,       lowerCamelCase_				)
			    and isinstance(lowerCamelCase_		,       lowerCamelCase_				)
			    and (numbera >= 0)
			    and (numbera >= 0)
			), "'number1' and 'number2' must been positive integer."

			__lowercase										=							0

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

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

			return numbera




def    _lowerCAmelCase						(				lowerCamelCase_			:  Optional[int]		,       lowerCamelCase_			:  List[Any]				):
			assert (
			    isinstance(lowerCamelCase_		,       lowerCamelCase_				)
			    and isinstance(lowerCamelCase_		,       lowerCamelCase_				)
			    and (numbera >= 1)
			    and (numbera >= 1)
			), "'number1' and 'number2' must been positive integer."

			__lowercase										=							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'
						__lowercase										=							prime_factorization(lowerCamelCase_				)
						__lowercase										=							prime_factorization(lowerCamelCase_				)

			elif numbera == 1 or numbera == 1:
						__lowercase										=							[]
						__lowercase										=							[]
						__lowercase										=							max(lowerCamelCase_		,       lowerCamelCase_				)

			__lowercase										=							0
			__lowercase										=							0

			__lowercase										=							[]  # 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:
												__lowercase										=							prime_fac_a.count(lowerCamelCase_				)
												__lowercase										=							prime_fac_a.count(lowerCamelCase_				)

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

									else:
												__lowercase										=							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:
									__lowercase										=							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    _lowerCAmelCase						(				lowerCamelCase_			:  str				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (n >= 0), "'number' must been a positive int"

			__lowercase										=							0
			__lowercase										=							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    _lowerCAmelCase						(				lowerCamelCase_			:  Any		,       lowerCamelCase_			:  List[Any]				):
			assert (
			    is_prime(lowerCamelCase_				) and is_prime(lowerCamelCase_				) and (p_number_a < p_number_a)
			), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

			__lowercase										=							p_number_a + 1  # jump to the next number

			__lowercase										=							[]  # 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    _lowerCAmelCase						(				lowerCamelCase_			:  Any				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (n >= 1), "'n' must been int and >= 1"

			__lowercase										=							[]  # 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    _lowerCAmelCase						(				lowerCamelCase_			:  Tuple				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (
			    number > 1
			), "'number' must been an int and >= 1"

			__lowercase										=							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    _lowerCAmelCase						(				lowerCamelCase_			:  int		,       lowerCamelCase_			:  List[str]				):
			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.
			__lowercase										=							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    _lowerCAmelCase						(				lowerCamelCase_			:  List[Any]				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (n >= 0), "'n' must been a int and >= 0"

			__lowercase										=							1  # this will be return.

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

			return ans




def    _lowerCAmelCase						(				lowerCamelCase_			:  Optional[Any]				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (n >= 0), "'n' must been an int and >= 0"

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

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

			return ans












'''simple docstring'''



import math




def    _lowerCAmelCase						(				lowerCamelCase_			:  int				):
			assert isinstance(lowerCamelCase_		,       lowerCamelCase_				) and (
			    number >= 0
			), "'number' must been an int and positive"

			if 1 < number < 4:
						# 2 and 3 are primes
						return True
			elif number < 2 or not number % 2:
						# Negatives, 0, 1 and all even numbers are not primes
						return False

			__lowercase										=							range(3		,       int(math.sqrt(lowerCamelCase_				) + 1				)		,       2				)
			return not any(not number % i for i in odd_numbers				)




def    _lowerCAmelCase						(				lowerCamelCase_			:  Dict		,       lowerCamelCase_			:  Any=1		,       **lowerCamelCase_			:  Tuple				):
			__lowercase										=							factor * value
			__lowercase										=							value

			while not is_prime(lowerCamelCase_				):
						value += 1 if not ("desc" in kwargs and kwargs["desc"] is True) else -1

			if value == first_value_val:
						return next_prime(value + 1		,       **lowerCamelCase_				)
			return value





