

from math import sqrt

def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  str           =     True

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

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

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

      return status

def    A_  (							a  ):

      """simple docstring"""



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

      # beginList: contains all natural numbers from 2 up to N
      SCREAMING_SNAKE_CASE_				:  Optional[int]           =     list(range(2					,   n + 1  )  )

      SCREAMING_SNAKE_CASE_				:  Any           =     []  # this list will be returns.

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

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

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

      return ans

def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  int           =     []

      # 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(a  ):
                  ans.append(a  )

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

      return ans

def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  Dict           =     []  # this list will be returns of the function.

      # potential prime number factors.

      SCREAMING_SNAKE_CASE_				:  List[Any]           =     2

      SCREAMING_SNAKE_CASE_				:  str           =     number

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

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

      else:
            ans.append(a  )

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

      return ans

def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  List[Any]           =     0

      # prime factorization of 'number'
      SCREAMING_SNAKE_CASE_				:  Tuple           =     prime_factorization(a  )

      SCREAMING_SNAKE_CASE_				:  Tuple           =     max(a  )

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

      return ans

def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  int           =     0

      # prime factorization of 'number'
      SCREAMING_SNAKE_CASE_				:  Any           =     prime_factorization(a  )

      SCREAMING_SNAKE_CASE_				:  Union[str, Any]           =     min(a  )

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

      return ans

def    A_  (							a  ):

      """simple docstring"""



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

      return number % 2 == 0

def    A_  (							a  ):

      """simple docstring"""



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

      return number % 2 != 0

def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  str           =     []  # this list will returned

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

      # run variable for while-loops.
      SCREAMING_SNAKE_CASE_				:  Tuple           =     0
      SCREAMING_SNAKE_CASE_				:  Optional[Any]           =     None

      # exit variable. for break up the loops
      SCREAMING_SNAKE_CASE_				:  str           =     True

      while i < len_pn and loop:
            SCREAMING_SNAKE_CASE_				:  Tuple           =     i + 1

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

                  j += 1

            i += 1

      # precondition
      assert (
          isinstance(a					,   a  )
          and (len(a  ) == 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_  (							a					,   a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  Dict           =     0

      while numbera != 0:
            SCREAMING_SNAKE_CASE_				:  List[Any]           =     numbera % numbera
            SCREAMING_SNAKE_CASE_				:  List[str]           =     numbera
            SCREAMING_SNAKE_CASE_				:  List[str]           =     rest

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

      return numbera

def    A_  (							a					,   a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  Dict           =     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_				:  Union[str, Any]           =     prime_factorization(a  )
            SCREAMING_SNAKE_CASE_				:  Tuple           =     prime_factorization(a  )

      elif numbera == 1 or numbera == 1:
            SCREAMING_SNAKE_CASE_				:  int           =     []
            SCREAMING_SNAKE_CASE_				:  Optional[int]           =     []
            SCREAMING_SNAKE_CASE_				:  Union[str, Any]           =     max(a					,   a  )

      SCREAMING_SNAKE_CASE_				:  List[Any]           =     0
      SCREAMING_SNAKE_CASE_				:  Union[str, Any]           =     0

      SCREAMING_SNAKE_CASE_				:  List[str]           =     []  # 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_				:  Dict           =     prime_fac_a.count(a  )
                        SCREAMING_SNAKE_CASE_				:  Tuple           =     prime_fac_a.count(a  )

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

                  else:
                        SCREAMING_SNAKE_CASE_				:  Optional[int]           =     prime_fac_a.count(a  )

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

                  done.append(a  )

    # iterates through primeFac2
      for n in prime_fac_a:
            if n not in done:
                  SCREAMING_SNAKE_CASE_				:  str           =     prime_fac_a.count(a  )

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

                  done.append(a  )

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

      return ans

def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  List[Any]           =     0
      SCREAMING_SNAKE_CASE_				:  Dict           =     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(a  ):
                  ans += 1

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

      return ans

def    A_  (							a					,   a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  Optional[int]           =     p_number_a + 1  # jump to the next number

      SCREAMING_SNAKE_CASE_				:  int           =     []  # this list will be returns.

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

      while number < p_number_a:
            ans.append(a  )

            number += 1

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

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

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

def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  Tuple           =     []  # will be returned.

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

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

      return ans

def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  Optional[Any]           =     get_divisors(a  )

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

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

def    A_  (							a					,   a  ):

      """simple docstring"""



      assert (
          isinstance(a					,   a  )
          and isinstance(a					,   a  )
          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_				:  str           =     gcd(abs(a  )					,   abs(a  )  )

      # precondition
      assert (
          isinstance(a					,   a  )
          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_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  Optional[int]           =     1  # this will be return.

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

      return ans



def    A_  (							a  ):

      """simple docstring"""



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

      SCREAMING_SNAKE_CASE_				:  Optional[int]           =     0
      SCREAMING_SNAKE_CASE_				:  Optional[int]           =     1
      SCREAMING_SNAKE_CASE_				:  Tuple           =     1  # this will be return

      for _ in range(n - 1  ):
            SCREAMING_SNAKE_CASE_				:  Optional[int]           =     ans
            ans += fiba
            SCREAMING_SNAKE_CASE_				:  Tuple           =     tmp

      return ans









def    A_  (							a  ):

      """simple docstring"""



      SCREAMING_SNAKE_CASE_				:  List[Any]           =     0
      while num > 0:
            digit_sum += num % 1_0
            num //= 1_0
      return digit_sum

def    A_  (							a = 1_0_0  ):

      """simple docstring"""



      SCREAMING_SNAKE_CASE_				:  List[str]           =     1
      SCREAMING_SNAKE_CASE_				:  Union[str, Any]           =     2

      for i in range(2					,   max_n + 1  ):
            SCREAMING_SNAKE_CASE_				:  List[str]           =     pre_numerator
            SCREAMING_SNAKE_CASE_				:  str           =     2 * i // 3 if i % 3 == 0 else 1
            SCREAMING_SNAKE_CASE_				:  Tuple           =     cur_numerator
            SCREAMING_SNAKE_CASE_				:  Tuple           =     e_cont * pre_numerator + temp

      return sum_digits(a  )


if __name__ == "__main__":
    print(F'{solution() = }')






