


'''simple docstring'''




from math import sqrt





def 	a_					( lowerCamelCase	:     int				):
   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_					( lowerCamelCase	:     Tuple				):
   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_					( lowerCamelCase	:     Optional[Any]				):
   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_					( lowerCamelCase	:     Dict				):
   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_					( lowerCamelCase	:     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     =			max(lowerCamelCase				)

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

   return ans





def 	a_					( lowerCamelCase	:     Optional[int]				):
   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_					( lowerCamelCase	:     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_					( lowerCamelCase	:     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_					( lowerCamelCase	:     Union[str, Any]				):
   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_					( lowerCamelCase	:     Dict	, lowerCamelCase	:     int				):
   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_					( lowerCamelCase	:     str	, lowerCamelCase	:     str				):
   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_					( lowerCamelCase	:     int				):
   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_					( lowerCamelCase	:     Optional[Any]	, lowerCamelCase	:     Optional[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_					( lowerCamelCase	:     Optional[Any]				):
   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_					( lowerCamelCase	:     List[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_					( lowerCamelCase	:     Optional[int]	, lowerCamelCase	:     Optional[int]				):
   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_					( lowerCamelCase	:     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_					( lowerCamelCase	:     List[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













from __future__ import annotations

UpperCAmelCase__				   =		list[list[int]]

# assigning initial values to the grid
UpperCAmelCase__				   =		[
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
UpperCAmelCase__				   =		[
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]
def 						_a    (      a :Matrix	,							a :int	,							a :int	,							a :int )							->			bool:
 for i in range(9 ):
  if grid[row][i] == n or grid[i][column] == n:
   return False

 for i in range(3 ):
  for j in range(3 ):
   if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
    return False

 return True
def 						_a    (      a :Matrix )							->			tuple[int, int] | None:
 for i in range(9 ):
  for j in range(9 ):
   if grid[i][j] == 0:
    return i, j
 return None
def 						_a    (      a :Matrix )							->			Matrix | None:
 if location := find_empty_location(a ):
  a  ,     a		  =      location
 else:
  # If the location is ``None``, then the grid is solved.
  return grid

 for digit in range(1	,							10 ):
  if is_safe(a	,							a	,							a	,							a ):
   a		  =      digit

   if sudoku(a ) is not None:
    return grid

   a		  =      0

 return None
def 						_a    (      a :Matrix )							->			None:
 for row in grid:
  for cell in row:
   print(a	,							end=''' ''' )
  print()


if __name__ == "__main__":
     # make a copy of grid so that you can compare with the unmodified grid
     for example_grid in (initial_grid, no_solution):
          print("\nExample grid:\n" + "=" * 20)
          print_solution(example_grid)
          print("\nExample grid solution:")
          UpperCAmelCase__				   =		sudoku(example_grid)
          if solution is not None:
               print_solution(solution)
          else:
               print("Cannot find a solution.")


