


"""simple docstring"""







from math import sqrt




def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


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

					return status




def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


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

					return ans




def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


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

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

					return ans




def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


					assert isinstance(lowerCAmelCase_  ,						lowerCAmelCase_   ) 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(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    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


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

					__SCREAMING_SNAKE_CASE						   =				0

					# prime factorization of 'number'
					__SCREAMING_SNAKE_CASE						   =				prime_factorization(lowerCAmelCase_   )

					__SCREAMING_SNAKE_CASE						   =				max(lowerCAmelCase_   )

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

					return ans




def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


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

					__SCREAMING_SNAKE_CASE						   =				0

					# prime factorization of 'number'
					__SCREAMING_SNAKE_CASE						   =				prime_factorization(lowerCAmelCase_   )

					__SCREAMING_SNAKE_CASE						   =				min(lowerCAmelCase_   )

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

					return ans




def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


					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    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


					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    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


					assert (
					    isinstance(lowerCAmelCase_  ,						lowerCAmelCase_   ) and (number > 2) and is_even(lowerCAmelCase_   )
					), "'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(lowerCAmelCase_   )
					__SCREAMING_SNAKE_CASE						   =				len(lowerCAmelCase_   )

					# 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(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    UpperCAmelCase__       (lowerCAmelCase_  ,						lowerCAmelCase_   ):




					'''simple docstring'''


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

					return numbera




def    UpperCAmelCase__       (lowerCAmelCase_  ,						lowerCAmelCase_   ):




					'''simple docstring'''


					assert (
					    isinstance(lowerCAmelCase_  ,						lowerCAmelCase_   )
					    and isinstance(lowerCAmelCase_  ,						lowerCAmelCase_   )
					    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(lowerCAmelCase_   )
										__SCREAMING_SNAKE_CASE						   =				prime_factorization(lowerCAmelCase_   )

					elif numbera == 1 or numbera == 1:
										__SCREAMING_SNAKE_CASE						   =				[]
										__SCREAMING_SNAKE_CASE						   =				[]
										__SCREAMING_SNAKE_CASE						   =				max(lowerCAmelCase_  ,						lowerCAmelCase_   )

					__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(lowerCAmelCase_   )
																				__SCREAMING_SNAKE_CASE						   =				prime_fac_a.count(lowerCAmelCase_   )

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

															else:
																				__SCREAMING_SNAKE_CASE						   =				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:
															__SCREAMING_SNAKE_CASE						   =				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    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


					assert isinstance(lowerCAmelCase_  ,						lowerCAmelCase_   ) 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(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    UpperCAmelCase__       (lowerCAmelCase_  ,						lowerCAmelCase_   ):




					'''simple docstring'''


					assert (
					    is_prime(lowerCAmelCase_   ) and is_prime(lowerCAmelCase_   ) 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(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    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


					assert isinstance(lowerCAmelCase_  ,						lowerCAmelCase_   ) 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(lowerCAmelCase_   )

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

					return ans




def    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


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

					__SCREAMING_SNAKE_CASE						   =				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    UpperCAmelCase__       (lowerCAmelCase_  ,						lowerCAmelCase_   ):




					'''simple docstring'''


					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.
					__SCREAMING_SNAKE_CASE						   =				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    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


					assert isinstance(lowerCAmelCase_  ,						lowerCAmelCase_   ) 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    UpperCAmelCase__       (lowerCAmelCase_   ):




					'''simple docstring'''


					assert isinstance(lowerCAmelCase_  ,						lowerCAmelCase_   ) 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"""







from __future__ import annotations




def    UpperCAmelCase__       (lowerCAmelCase_  ,						lowerCAmelCase_ = None  ,						lowerCAmelCase_ = None   ):




					'''simple docstring'''


					if start is None:
										__SCREAMING_SNAKE_CASE						   =				0

					if end is None:
										__SCREAMING_SNAKE_CASE						   =				len(lowerCAmelCase_   ) - 1

					if start >= end:
										return

					__SCREAMING_SNAKE_CASE						   =				(start + end) // 2

					slowsort(lowerCAmelCase_  ,						lowerCAmelCase_  ,						lowerCAmelCase_   )
					slowsort(lowerCAmelCase_  ,						mid + 1  ,						lowerCAmelCase_   )

					if sequence[end] < sequence[mid]:
										__SCREAMING_SNAKE_CASE   ,  __SCREAMING_SNAKE_CASE						   =				sequence[mid], sequence[end]

					slowsort(lowerCAmelCase_  ,						lowerCAmelCase_  ,						end - 1   )


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

		testmod()
