


"""simple docstring"""







from math import sqrt
def 					a__			(				SCREAMING_SNAKE_CASE    :    int				):


  '''simple docstring'''







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

  lowerCAmelCase     :     Any								=							True

  # 0 and 1 are none primes.
  if number <= 1:
    lowerCAmelCase     :     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     :     Optional[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    :    str				):


  '''simple docstring'''







  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     :     Tuple								=							list(range(2				,      n + 1				)				)

  lowerCAmelCase     :     List[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     :     Tuple								=							0

    # filters actual prime numbers.
  lowerCAmelCase     :     str								=							[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				):


  '''simple docstring'''







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

  lowerCAmelCase     :     str								=							[]

  # 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    :    Optional[Any]				):


  '''simple docstring'''







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

  lowerCAmelCase     :     Optional[Any]								=							[]  # this list will be returns of the function.

  # potential prime number factors.

  lowerCAmelCase     :     Optional[Any]								=							2

  lowerCAmelCase     :     Optional[Any]								=							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    :    str				):


  '''simple docstring'''







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

  lowerCAmelCase     :     List[Any]								=							0

  # prime factorization of 'number'
  lowerCAmelCase     :     Union[str, Any]								=							prime_factorization(SCREAMING_SNAKE_CASE				)

  lowerCAmelCase     :     Any								=							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    :    List[Any]				):


  '''simple docstring'''







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

  lowerCAmelCase     :     Any								=							0

  # prime factorization of 'number'
  lowerCAmelCase     :     str								=							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    :    Dict				):


  '''simple docstring'''







  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    :    int				):


  '''simple docstring'''







  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    :    Optional[int]				):


  '''simple docstring'''







  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     :     str								=							[]  # this list will returned

  # creates a list of prime numbers between 2 up to 'number'
  lowerCAmelCase     :     Any								=							get_prime_numbers(SCREAMING_SNAKE_CASE				)
  lowerCAmelCase     :     Optional[int]								=							len(SCREAMING_SNAKE_CASE				)

  # run variable for while-loops.
  lowerCAmelCase     :     Dict								=							0
  lowerCAmelCase     :     Optional[int]								=							None

  # exit variable. for break up the loops
  lowerCAmelCase     :     Optional[int]								=							True

  while i < len_pn and loop:
    lowerCAmelCase     :     int								=							i + 1

    while j < len_pn and loop:
      if prime_numbers[i] + prime_numbers[j] == number:
        lowerCAmelCase     :     Dict								=							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    :    Tuple				,      SCREAMING_SNAKE_CASE    :    Any				):


  '''simple docstring'''







  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     :     int								=							0

  while numbera != 0:
    lowerCAmelCase     :     Dict								=							numbera % numbera
    lowerCAmelCase     :     int								=							numbera
    lowerCAmelCase     :     Optional[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    :    Optional[int]				,      SCREAMING_SNAKE_CASE    :    List[str]				):


  '''simple docstring'''







  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     :     List[str]								=							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     :     Dict								=							prime_factorization(SCREAMING_SNAKE_CASE				)
    lowerCAmelCase     :     Optional[int]								=							prime_factorization(SCREAMING_SNAKE_CASE				)

  elif numbera == 1 or numbera == 1:
    lowerCAmelCase     :     Any								=							[]
    lowerCAmelCase     :     Tuple								=							[]
    lowerCAmelCase     :     List[str]								=							max(SCREAMING_SNAKE_CASE				,      SCREAMING_SNAKE_CASE				)

  lowerCAmelCase     :     Optional[int]								=							0
  lowerCAmelCase     :     Optional[Any]								=							0

  lowerCAmelCase     :     Union[str, Any]								=							[]  # 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     :     Any								=							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     :     List[str]								=							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     :     Any								=							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				):


  '''simple docstring'''







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

  lowerCAmelCase     :     int								=							0
  lowerCAmelCase     :     Tuple								=							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    :    List[Any]				,      SCREAMING_SNAKE_CASE    :    Optional[Any]				):


  '''simple docstring'''







  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     :     List[Any]								=							p_number_a + 1  # jump to the next number

  lowerCAmelCase     :     Tuple								=							[]  # 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    :    Optional[int]				):


  '''simple docstring'''







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

  lowerCAmelCase     :     Any								=							[]  # 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    :    List[Any]				):


  '''simple docstring'''







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

  lowerCAmelCase     :     Any								=							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    :    List[str]				,      SCREAMING_SNAKE_CASE    :    Tuple				):


  '''simple docstring'''







  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     :     Any								=							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    :    Any				):


  '''simple docstring'''







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

  lowerCAmelCase     :     Any								=							1  # this will be return.

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

  return ans





def 					a__			(				SCREAMING_SNAKE_CASE    :    str				):


  '''simple docstring'''







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

  lowerCAmelCase     :     Union[str, Any]								=							0
  lowerCAmelCase     :     Tuple								=							1
  lowerCAmelCase     :     Any								=							1  # this will be return

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

  return ans









"""simple docstring"""







def 					a__			(				SCREAMING_SNAKE_CASE    :    int				,      SCREAMING_SNAKE_CASE    :    int				):


  '''simple docstring'''







  while b:
    lowerCAmelCase						,  lowerCAmelCase     :     Any								=							b, a % b
  return a
def 					a__			(				SCREAMING_SNAKE_CASE    :    int				,      SCREAMING_SNAKE_CASE    :    int				):


  '''simple docstring'''







  return a if b == 0 else euclidean_gcd_recursive(SCREAMING_SNAKE_CASE				,      a % b				)
def 					a__			(				):


  '''simple docstring'''







  print(f"""euclidean_gcd(3, 5) = {euclidean_gcd(3				,      5				)}"""				)
  print(f"""euclidean_gcd(5, 3) = {euclidean_gcd(5				,      3				)}"""				)
  print(f"""euclidean_gcd(1, 3) = {euclidean_gcd(1				,      3				)}"""				)
  print(f"""euclidean_gcd(3, 6) = {euclidean_gcd(3				,      6				)}"""				)
  print(f"""euclidean_gcd(6, 3) = {euclidean_gcd(6				,      3				)}"""				)

  print(f"""euclidean_gcd_recursive(3, 5) = {euclidean_gcd_recursive(3				,      5				)}"""				)
  print(f"""euclidean_gcd_recursive(5, 3) = {euclidean_gcd_recursive(5				,      3				)}"""				)
  print(f"""euclidean_gcd_recursive(1, 3) = {euclidean_gcd_recursive(1				,      3				)}"""				)
  print(f"""euclidean_gcd_recursive(3, 6) = {euclidean_gcd_recursive(3				,      6				)}"""				)
  print(f"""euclidean_gcd_recursive(6, 3) = {euclidean_gcd_recursive(6				,      3				)}"""				)


if __name__ == "__main__":
     main()





