






from math import sqrt




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					_A								= True

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

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

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

					return status




def      __A					(     _lowercase ):

					'''simple docstring'''







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

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

					_A								= []  # this list will be returns.

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

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

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

					return ans




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					_A								= []

					# 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(_lowercase ):
															ans.append(_lowercase )

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

					return ans




def      __A					(     _lowercase ):

					'''simple docstring'''







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

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

					# potential prime number factors.

					_A								= 2

					_A								= number

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

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

					else:
										ans.append(_lowercase )

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

					return ans




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					_A								= 0

					# prime factorization of 'number'
					_A								= prime_factorization(_lowercase )

					_A								= max(_lowercase )

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

					return ans




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					_A								= 0

					# prime factorization of 'number'
					_A								= prime_factorization(_lowercase )

					_A								= min(_lowercase )

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

					return ans




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					return number % 2 == 0




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					return number % 2 != 0




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					_A								= []  # this list will returned

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

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

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

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

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

															j += 1

										i += 1

					# precondition
					assert (
					    isinstance(_lowercase					,			_lowercase )
					    and (len(_lowercase ) == 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					(     _lowercase					,			_lowercase ):

					'''simple docstring'''







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

					_A								= 0

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

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

					return numbera




def      __A					(     _lowercase					,			_lowercase ):

					'''simple docstring'''







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

					_A								= 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'
										_A								= prime_factorization(_lowercase )
										_A								= prime_factorization(_lowercase )

					elif numbera == 1 or numbera == 1:
										_A								= []
										_A								= []
										_A								= max(_lowercase					,			_lowercase )

					_A								= 0
					_A								= 0

					_A								= []  # 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:
																				_A								= prime_fac_a.count(_lowercase )
																				_A								= prime_fac_a.count(_lowercase )

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

															else:
																				_A								= prime_fac_a.count(_lowercase )

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

															done.append(_lowercase )

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

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

															done.append(_lowercase )

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

					return ans




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					_A								= 0
					_A								= 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(_lowercase ):
															ans += 1

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

					return ans




def      __A					(     _lowercase					,			_lowercase ):

					'''simple docstring'''







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

					_A								= p_number_a + 1  # jump to the next number

					_A								= []  # this list will be returns.

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

					while number < p_number_a:
										ans.append(_lowercase )

										number += 1

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

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

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




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					_A								= []  # will be returned.

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

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

					return ans




def      __A					(     _lowercase ):

					'''simple docstring'''







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

					_A								= get_divisors(_lowercase )

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

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




def      __A					(     _lowercase					,			_lowercase ):

					'''simple docstring'''







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

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

					# precondition
					assert (
					    isinstance(_lowercase					,			_lowercase )
					    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					(     _lowercase ):

					'''simple docstring'''







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

					_A								= 1  # this will be return.

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

					return ans




def      __A					(     _lowercase ):

					'''simple docstring'''







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

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

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

					return ans











def      __A					(     _lowercase					,			_lowercase ):

					'''simple docstring'''







					_A								= word.split()

					def justify(_lowercase					,			_lowercase					,			_lowercase ) -> str:
										_A								= max_width - width
										_A								= len(_lowercase )
										if len(_lowercase ) == 1:
															# if there is only word in line
															# just insert overall_spaces_count for the remainder of line
															return line[0] + " " * overall_spaces_count
										else:
															_A								= words_count - 1
															# num_spaces_between_words_list[i] : tells you to insert
															# num_spaces_between_words_list[i] spaces
															# after word on line[i]
															_A								= spaces_to_insert_between_words * [
															    overall_spaces_count // spaces_to_insert_between_words
															]
															_A								= (
															    overall_spaces_count % spaces_to_insert_between_words
															)
															# distribute spaces via round robin to the left words
															for i in range(_lowercase ):
																				num_spaces_between_words_list[i] += 1
															_A								= []
															for i in range(_lowercase ):
																				# add the word
																				aligned_words_list.append(line[i] )
																				# add the spaces to insert
																				aligned_words_list.append(num_spaces_between_words_list[i] * ''' ''' )
															# just add the last word to the sentence
															aligned_words_list.append(line[-1] )
															# join the aligned words list to form a justified line
															return "".join(_lowercase )

					_A								= []
					_A								= []
					_A								= 0
					for word in words:
										if width + len(_lowercase ) + len(_lowercase ) <= max_width:
															# keep adding words until we can fill out max_width
															# width = sum of length of all words (without overall_spaces_count)
															# len(word) = length of current word
															# len(line) = number of overall_spaces_count to insert between words
															line.append(_lowercase )
															width += len(_lowercase )
										else:
															# justify the line and add it to result
															answer.append(justify(_lowercase					,			_lowercase					,			_lowercase ) )
															# reset new line and new width
															_A							,_A								= [word], len(_lowercase )
					_A								= max_width - width - len(_lowercase )
					answer.append(''' '''.join(_lowercase ) + (remaining_spaces + 1) * ''' ''' )
					return answer


if __name__ == "__main__":
				from doctest import testmod

				testmod()



