
def 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    list[list]:
       lowercase							:     Dict       	=      current_set.copy()
       for row_index, row in enumerate(SCREAMING_SNAKE_CASE__				):
              lowercase							:     Tuple       	=      row[0]
              for column_index, column in enumerate(SCREAMING_SNAKE_CASE__				):
                     if magnitude == 0:
                            lowercase							:     Union[str, Any]       	=      column
                            continue
                     lowercase							:     List[Any]       	=      column / magnitude
    # Subtract to cancel term
       lowercase							:     Tuple       	=      current_set[0]
       lowercase							:     Tuple       	=      [first_row]
       lowercase							:     Union[str, Any]       	=      current_set[1::]
       for row in current_set:
              lowercase							:     Any       	=      []
              # If first term is 0, it is already in form we want, so we preserve it
              if row[0] == 0:
                     final_set.append(SCREAMING_SNAKE_CASE__				)
                     continue
              for column_index in range(len(SCREAMING_SNAKE_CASE__				)				):
                     temp_row.append(first_row[column_index] - row[column_index]				)
              final_set.append(SCREAMING_SNAKE_CASE__				)
       # Create next recursion iteration set
       if len(final_set[0]				) != 3:
              lowercase							:     List[str]       	=      final_set[0]
              lowercase							:     Dict       	=      []
              lowercase							:     str       	=      []
              for row in final_set[1::]:
                     current_first_column.append(row[0]				)
                     next_iteration.append(row[1::]				)
              lowercase							:     Dict       	=      simplify(SCREAMING_SNAKE_CASE__				)
              for i in range(len(SCREAMING_SNAKE_CASE__				)				):
                     resultant[i].insert(0					,	current_first_column[i]				)
              resultant.insert(0					,	SCREAMING_SNAKE_CASE__				)
              lowercase							:     Any       	=      resultant
       return final_set







def 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    list:
       if len(SCREAMING_SNAKE_CASE__				) == 0:
              raise IndexError("""solve_simultaneous() requires n lists of length n+1"""				)
       lowercase							:     str       	=      len(SCREAMING_SNAKE_CASE__				) + 1
       if any(len(SCREAMING_SNAKE_CASE__				) != _length for item in equations				):
              raise IndexError("""solve_simultaneous() requires n lists of length n+1"""				)
       for row in equations:
              if any(not isinstance(SCREAMING_SNAKE_CASE__					,	(int, float)				) for column in row				):
                     raise ValueError("""solve_simultaneous() requires lists of integers"""				)
       if len(SCREAMING_SNAKE_CASE__				) == 1:
              return [equations[0][-1] / equations[0][0]]
       lowercase							:     str       	=      equations.copy()
       if any(0 in row for row in data_set				):
              lowercase							:     Dict       	=      data_set.copy()
              lowercase							:     int       	=      []
              for row_index, row in enumerate(SCREAMING_SNAKE_CASE__				):
                     if 0 not in row:
                            lowercase							:     Tuple       	=      data_set.pop(SCREAMING_SNAKE_CASE__				)
                            break
              if not full_row:
                     raise ValueError("""solve_simultaneous() requires at least 1 full equation"""				)
              data_set.insert(0					,	SCREAMING_SNAKE_CASE__				)
       lowercase							:     Any       	=      data_set.copy()
       lowercase							:     List[str]       	=      simplify(SCREAMING_SNAKE_CASE__				)
       lowercase							:     int       	=      simplified[::-1]
       lowercase							:     list       	=      []
       for row in simplified:
              lowercase							:     List[Any]       	=      row[-1]
              if not solutions:
                     if row[-2] == 0:
                            solutions.append(0				)
                            continue
                     solutions.append(current_solution / row[-2]				)
                     continue
              lowercase							:     int       	=      row.copy()[: len(SCREAMING_SNAKE_CASE__				) - 1 :]
              while temp_row[0] == 0:
                     temp_row.pop(0				)
              if len(SCREAMING_SNAKE_CASE__				) == 0:
                     solutions.append(0				)
                     continue
              lowercase							:     Optional[Any]       	=      temp_row[1::]
              lowercase							:     Dict       	=      temp_row[::-1]
              for column_index, column in enumerate(SCREAMING_SNAKE_CASE__				):
                     current_solution -= column * solutions[column_index]
              solutions.append(SCREAMING_SNAKE_CASE__				)
       lowercase							:     Any       	=      []
       for item in solutions:
              final.append(float(round(SCREAMING_SNAKE_CASE__					,	5				)				)				)
       return final[::-1]


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  lowercase  :					Any						    =							[
      [2, 1, 1, 1, 1, 4],
      [1, 2, 1, 1, 1, 5],
      [1, 1, 2, 1, 1, 6],
      [1, 1, 1, 2, 1, 7],
      [1, 1, 1, 1, 2, 8],
  ]
  print(solve_simultaneous(eq))
  print(solve_simultaneous([[4, 2]]))


from math import sqrt







def 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    bool:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and (
           number >= 0
       ), "'number' must been an int and positive"

       lowercase							:     Union[str, Any]       	=      True

       # 0 and 1 are none primes.
       if number <= 1:
              lowercase							:     str       	=      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:
                     lowercase							:     Any       	=      False
                     break

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

       return status







def 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    Union[str, Any]:
       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
       lowercase							:     str       	=      list(range(2					,	n + 1				)				)

       lowercase							:     Tuple       	=      []  # 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):
                            lowercase							:     Tuple       	=      0

    # filters actual prime numbers.
       lowercase							:     int       	=      [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 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    Optional[Any]:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and (n > 2), "'N' must been an int and > 2"

       lowercase							:     Dict       	=      []

       # 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 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    Tuple:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and number >= 0, "'number' must been an int and >= 0"

       lowercase							:     Tuple       	=      []  # this list will be returns of the function.

       # potential prime number factors.

       lowercase							:     Optional[Any]       	=      2

       lowercase							:     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 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    List[Any]:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and (
           number >= 0
       ), "'number' bust been an int and >= 0"

       lowercase							:     Tuple       	=      0

       # prime factorization of 'number'
       lowercase							:     Optional[int]       	=      prime_factorization(SCREAMING_SNAKE_CASE__				)

       lowercase							:     Union[str, Any]       	=      max(SCREAMING_SNAKE_CASE__				)

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

       return ans







def 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    str:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and (
           number >= 0
       ), "'number' bust been an int and >= 0"

       lowercase							:     Union[str, Any]       	=      0

       # prime factorization of 'number'
       lowercase							:     Tuple       	=      prime_factorization(SCREAMING_SNAKE_CASE__				)

       lowercase							:     Union[str, Any]       	=      min(SCREAMING_SNAKE_CASE__				)

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

       return ans







def 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    Optional[Any]:
       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 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    int:
       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 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    int:
       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"

       lowercase							:     Union[str, Any]       	=      []  # this list will returned

       # creates a list of prime numbers between 2 up to 'number'
       lowercase							:     str       	=      get_prime_numbers(SCREAMING_SNAKE_CASE__				)
       lowercase							:     Any       	=      len(SCREAMING_SNAKE_CASE__				)

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

       # exit variable. for break up the loops
       lowercase							:     Any       	=      True

       while i < len_pn and loop:
              lowercase							:     str       	=      i + 1

              while j < len_pn and loop:
                     if prime_numbers[i] + prime_numbers[j] == number:
                            lowercase							:     Union[str, Any]       	=      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 						_snake_case(  SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) ->    str:
       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."

       lowercase							:     Union[str, Any]       	=      0

       while numbera != 0:
              lowercase							:     Optional[int]       	=      numbera % numbera
              lowercase							:     Optional[int]       	=      numbera
              lowercase							:     Dict       	=      rest

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

       return numbera







def 						_snake_case(  SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) ->    Tuple:
       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."

       lowercase							:     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'
              lowercase							:     Optional[Any]       	=      prime_factorization(SCREAMING_SNAKE_CASE__				)
              lowercase							:     Optional[int]       	=      prime_factorization(SCREAMING_SNAKE_CASE__				)

       elif numbera == 1 or numbera == 1:
              lowercase							:     Union[str, Any]       	=      []
              lowercase							:     List[str]       	=      []
              lowercase							:     Dict       	=      max(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				)

       lowercase							:     Union[str, Any]       	=      0
       lowercase							:     Optional[Any]       	=      0

       lowercase							:     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:
                            lowercase							:     Dict       	=      prime_fac_a.count(SCREAMING_SNAKE_CASE__				)
                            lowercase							:     Optional[Any]       	=      prime_fac_a.count(SCREAMING_SNAKE_CASE__				)

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

                     else:
                            lowercase							:     List[Any]       	=      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:
                     lowercase							:     Optional[int]       	=      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 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    Any:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and (n >= 0), "'number' must been a positive int"

       lowercase							:     Dict       	=      0
       lowercase							:     List[str]       	=      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 						_snake_case(  SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) ->    List[Any]:
       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'"

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

       lowercase							:     List[Any]       	=      []  # 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 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    Any:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and (n >= 1), "'n' must been int and >= 1"

       lowercase							:     Optional[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 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    Any:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and (
           number > 1
       ), "'number' must been an int and >= 1"

       lowercase							:     str       	=      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 						_snake_case(  SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) ->    Dict:
       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.
       lowercase							:     Tuple       	=      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 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    int:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and (n >= 0), "'n' must been a int and >= 0"

       lowercase							:     List[str]       	=      1  # this will be return.

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

       return ans







def 						_snake_case(  SCREAMING_SNAKE_CASE__				) ->    Union[str, Any]:
       assert isinstance(SCREAMING_SNAKE_CASE__					,	SCREAMING_SNAKE_CASE__				) and (n >= 0), "'n' must been an int and >= 0"

       lowercase							:     int       	=      0
       lowercase							:     Union[str, Any]       	=      1
       lowercase							:     int       	=      1  # this will be return

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

       return ans
