




def 					_A	(		SCREAMING_SNAKE_CASE					:       list[list]						):





  """simple docstring"""



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





  """simple docstring"""



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


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     UpperCAmelCase     :     List[str]							=	[
         [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]]))













def 					_A	(		SCREAMING_SNAKE_CASE					:       int = 1	,	SCREAMING_SNAKE_CASE					:       int = 1_000						):





  """simple docstring"""



  a__     : Any  =1
  a__     : Any  =0

  for divide_by_number in range(SCREAMING_SNAKE_CASE	,	digit + 1						):
    a__     : list[int]  =[]
    a__     : int  =numerator
    for _ in range(1	,	digit + 1						):
      if now_divide in has_been_divided:
        if longest_list_length < len(SCREAMING_SNAKE_CASE						):
          a__     : Optional[Any]  =len(SCREAMING_SNAKE_CASE						)
          a__     : List[str]  =divide_by_number
      else:
        has_been_divided.append(SCREAMING_SNAKE_CASE						)
        a__     : List[Any]  =now_divide * 10 % divide_by_number

  return the_digit


# Tests
if __name__ == "__main__":
     import doctest

     doctest.testmod()






