




from math import sqrt







def 			_a   (		SCREAMING_SNAKE_CASE_							:       int  )     ->   List[str]:
					assert isinstance(lowerCamelCase_   ,   lowerCamelCase_  ) and (
					    number >= 0
					), "'number' must been an int and positive"

					__lowerCAmelCase            = True

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

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

					return status







def 			_a   (		SCREAMING_SNAKE_CASE_							:       Optional[Any]  )     ->   List[str]:
					assert isinstance(lowerCamelCase_   ,   lowerCamelCase_  ) and (n > 2), "'N' must been an int and > 2"

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

					__lowerCAmelCase            = []  # 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):
																				__lowerCAmelCase            = 0

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

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

					return ans







def 			_a   (		SCREAMING_SNAKE_CASE_							:       Any  )     ->   List[str]:
					assert isinstance(lowerCamelCase_   ,   lowerCamelCase_  ) and (n > 2), "'N' must been an int and > 2"

					__lowerCAmelCase            = []

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

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

					# potential prime number factors.

					__lowerCAmelCase            = 2

					__lowerCAmelCase            = 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 			_a   (		SCREAMING_SNAKE_CASE_							:       Any  )     ->   Union[str, Any]:
					assert isinstance(lowerCamelCase_   ,   lowerCamelCase_  ) and (
					    number >= 0
					), "'number' bust been an int and >= 0"

					__lowerCAmelCase            = 0

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

					__lowerCAmelCase            = max(lowerCamelCase_  )

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

					return ans







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

					__lowerCAmelCase            = 0

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

					__lowerCAmelCase            = min(lowerCamelCase_  )

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

					return ans







def 			_a   (		SCREAMING_SNAKE_CASE_							:       Tuple  )     ->   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 			_a   (		SCREAMING_SNAKE_CASE_							:       Optional[Any]  )     ->   Union[str, 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 			_a   (		SCREAMING_SNAKE_CASE_							:       int  )     ->   List[str]:
					assert (
					    isinstance(lowerCamelCase_   ,   lowerCamelCase_  ) and (number > 2) and is_even(lowerCamelCase_  )
					), "'number' must been an int, even and > 2"

					__lowerCAmelCase            = []  # this list will returned

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

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

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

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

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

					__lowerCAmelCase            = 0

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

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

					return numbera







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

					__lowerCAmelCase            = 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            = prime_factorization(lowerCamelCase_  )
										__lowerCAmelCase            = prime_factorization(lowerCamelCase_  )

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

					__lowerCAmelCase            = 0
					__lowerCAmelCase            = 0

					__lowerCAmelCase            = []  # 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            = prime_fac_a.count(lowerCamelCase_  )
																				__lowerCAmelCase            = prime_fac_a.count(lowerCamelCase_  )

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

															else:
																				__lowerCAmelCase            = 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:
															__lowerCAmelCase            = 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 			_a   (		SCREAMING_SNAKE_CASE_							:       str  )     ->   Tuple:
					assert isinstance(lowerCamelCase_   ,   lowerCamelCase_  ) and (n >= 0), "'number' must been a positive int"

					__lowerCAmelCase            = 0
					__lowerCAmelCase            = 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 			_a   (		SCREAMING_SNAKE_CASE_							:       Any   ,   SCREAMING_SNAKE_CASE_							:       List[Any]  )     ->   int:
					assert (
					    is_prime(lowerCamelCase_  ) and is_prime(lowerCamelCase_  ) and (p_number_a < p_number_a)
					), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

					__lowerCAmelCase            = p_number_a + 1  # jump to the next number

					__lowerCAmelCase            = []  # 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 			_a   (		SCREAMING_SNAKE_CASE_							:       Any  )     ->   int:
					assert isinstance(lowerCamelCase_   ,   lowerCamelCase_  ) and (n >= 1), "'n' must been int and >= 1"

					__lowerCAmelCase            = []  # 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 			_a   (		SCREAMING_SNAKE_CASE_							:       Tuple  )     ->   Optional[Any]:
					assert isinstance(lowerCamelCase_   ,   lowerCamelCase_  ) and (
					    number > 1
					), "'number' must been an int and >= 1"

					__lowerCAmelCase            = 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 			_a   (		SCREAMING_SNAKE_CASE_							:       int   ,   SCREAMING_SNAKE_CASE_							:       List[str]  )     ->   Optional[Any]:
					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.
					__lowerCAmelCase            = 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 			_a   (		SCREAMING_SNAKE_CASE_							:       List[Any]  )     ->   List[str]:
					assert isinstance(lowerCamelCase_   ,   lowerCamelCase_  ) and (n >= 0), "'n' must been a int and >= 0"

					__lowerCAmelCase            = 1  # this will be return.

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

					return ans







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

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

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

					return ans










import os


# All paths are set with the intent you should run this script from the root of the repo with the command
# python utils/check_doctest_list.py
UpperCamelCase__					    =    """."""


if __name__ == "__main__":
					UpperCamelCase__					    =    os.path.join(REPO_PATH, """utils/documentation_tests.txt""")
					UpperCamelCase__					    =    []
					UpperCamelCase__					    =    []
					with open(doctest_file_path) as fp:
										for line in fp:
															UpperCamelCase__					    =    line.strip()
															UpperCamelCase__					    =    os.path.join(REPO_PATH, line)
															if not (os.path.isfile(path) or os.path.isdir(path)):
																				non_existent_paths.append(line)
															all_paths.append(path)
					if len(non_existent_paths) > 0:
										UpperCamelCase__					    =    """\n""".join(non_existent_paths)
										raise ValueError(f'''`utils/documentation_tests.txt` contains non-existent paths:\n{non_existent_paths}''')
					if all_paths != sorted(all_paths):
										raise ValueError("""Files in `utils/documentation_tests.txt` are not in alphabetical order.""")




