def       lowerCAmelCase_     (		lowercase:							list[list]		)    ->				list[list]:







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




def       lowerCAmelCase_     (		lowercase:							list[list]		)    ->				list:







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


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  UpperCAmelCase_			    =							[
      [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       lowerCAmelCase_     (		lowercase:							int = 600_851_475_143		)    ->				int:







 '''simple docstring'''

 try:
  _UpperCamelCase:     Optional[int]							= int(lowercase		)
 except (TypeError, ValueError):
  raise TypeError('''Parameter n must be int or castable to int.'''		)
 if n <= 0:
  raise ValueError('''Parameter n must be greater than or equal to one.'''		)
 _UpperCamelCase:     Dict							= 1
 _UpperCamelCase:     Tuple							= 2
 while i * i <= n:
  while n % i == 0:
   _UpperCamelCase:     List[Any]							= i
   n //= i
  i += 1
 if n > 1:
  _UpperCamelCase:     Any							= n
 return int(lowercase		)


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