from math import sqrt


def      a__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) 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(__UpperCamelCase				)				)				) + 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(__UpperCamelCase     ,      __UpperCamelCase				), "'status' must been from type bool"

      return status


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

      return ans


def      a__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) 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(__UpperCamelCase				):
                  ans.append(__UpperCamelCase				)

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

      return ans


def      a__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) 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(__UpperCamelCase				)

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

      else:
            ans.append(__UpperCamelCase				)

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

      return ans


def      a__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) and (
          number >= 0
      ), "'number' bust been an int and >= 0"

      SCREAMING_SNAKE_CASE_          =  0

      # prime factorization of 'number'
      SCREAMING_SNAKE_CASE_          =  prime_factorization(__UpperCamelCase				)

      SCREAMING_SNAKE_CASE_          =  max(__UpperCamelCase				)

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

      return ans


def      a__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) and (
          number >= 0
      ), "'number' bust been an int and >= 0"

      SCREAMING_SNAKE_CASE_          =  0

      # prime factorization of 'number'
      SCREAMING_SNAKE_CASE_          =  prime_factorization(__UpperCamelCase				)

      SCREAMING_SNAKE_CASE_          =  min(__UpperCamelCase				)

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

      return ans


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

      return number % 2 == 0


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

      return number % 2 != 0


def      a__ (		__UpperCamelCase				):
      assert (
          isinstance(__UpperCamelCase     ,      __UpperCamelCase				) and (number > 2) and is_even(__UpperCamelCase				)
      ), "'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(__UpperCamelCase				)
      SCREAMING_SNAKE_CASE_          =  len(__UpperCamelCase				)

      # 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(__UpperCamelCase     ,      __UpperCamelCase				)
          and (len(__UpperCamelCase				) == 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__ (		__UpperCamelCase     ,      __UpperCamelCase				):
      assert (
          isinstance(__UpperCamelCase     ,      __UpperCamelCase				)
          and isinstance(__UpperCamelCase     ,      __UpperCamelCase				)
          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(__UpperCamelCase     ,      __UpperCamelCase				) and (
          numbera >= 0
      ), "'number' must been from type int and positive"

      return numbera


def      a__ (		__UpperCamelCase     ,      __UpperCamelCase				):
      assert (
          isinstance(__UpperCamelCase     ,      __UpperCamelCase				)
          and isinstance(__UpperCamelCase     ,      __UpperCamelCase				)
          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(__UpperCamelCase				)
            SCREAMING_SNAKE_CASE_          =  prime_factorization(__UpperCamelCase				)

      elif numbera == 1 or numbera == 1:
            SCREAMING_SNAKE_CASE_          =  []
            SCREAMING_SNAKE_CASE_          =  []
            SCREAMING_SNAKE_CASE_          =  max(__UpperCamelCase     ,      __UpperCamelCase				)

      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(__UpperCamelCase				)
                        SCREAMING_SNAKE_CASE_          =  prime_fac_a.count(__UpperCamelCase				)

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

                  else:
                        SCREAMING_SNAKE_CASE_          =  prime_fac_a.count(__UpperCamelCase				)

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

                  done.append(__UpperCamelCase				)

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

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

                  done.append(__UpperCamelCase				)

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

      return ans


def      a__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) 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(__UpperCamelCase				):
                  ans += 1

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

      return ans


def      a__ (		__UpperCamelCase     ,      __UpperCamelCase				):
      assert (
          is_prime(__UpperCamelCase				) and is_prime(__UpperCamelCase				) 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(__UpperCamelCase				):
            number += 1

      while number < p_number_a:
            ans.append(__UpperCamelCase				)

            number += 1

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

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

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


def      a__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) 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(__UpperCamelCase				)

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

      return ans


def      a__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) and (
          number > 1
      ), "'number' must been an int and >= 1"

      SCREAMING_SNAKE_CASE_          =  get_divisors(__UpperCamelCase				)

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

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


def      a__ (		__UpperCamelCase     ,      __UpperCamelCase				):
      assert (
          isinstance(__UpperCamelCase     ,      __UpperCamelCase				)
          and isinstance(__UpperCamelCase     ,      __UpperCamelCase				)
          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(__UpperCamelCase				)     ,      abs(__UpperCamelCase				)				)

      # precondition
      assert (
          isinstance(__UpperCamelCase     ,      __UpperCamelCase				)
          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__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) 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      a__ (		__UpperCamelCase				):
      assert isinstance(__UpperCamelCase     ,      __UpperCamelCase				) 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



from typing import Optional
from urllib.parse import quote

import huggingface_hub as hfh
from packaging import version


def      a__ (		__UpperCamelCase     ,      __UpperCamelCase     ,      __UpperCamelCase = None				):
      if version.parse(hfh.__version__				).release < version.parse("0.11.0"				).release:
            # old versions of hfh don't url-encode the file path
            SCREAMING_SNAKE_CASE_          =  quote(__UpperCamelCase				)
      return hfh.hf_hub_url(__UpperCamelCase     ,      __UpperCamelCase     ,      repo_type="dataset"     ,      revision=__UpperCamelCase				)


