



'''simple docstring'''


import math
from collections.abc import Iterator
from itertools import takewhile







def 				__lowerCamelCase				(   A__    )       ->	Optional[int]:






   """simple docstring"""







   if 1 < number < 4:
      # 2 and 3 are primes
      return True
   elif number < 2 or number % 2 == 0 or number % 3 == 0:
      # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
      return False

   # All primes number are in format of 6k +/- 1
   for i in range(5    ,						int(math.sqrt(__UpperCamelCase    ) + 1    )    ,						6    ):
      if number % i == 0 or number % (i + 2) == 0:
         return False
   return True







def 				__lowerCamelCase				(   )       ->	Optional[int]:






   """simple docstring"""
   UpperCamelCase					=  2
   while True:
      if is_prime(__UpperCamelCase    ):
         yield num
      num += 1







def 				__lowerCamelCase				(   A__ = 2_000_000    )       ->	List[str]:






   """simple docstring"""
   return sum(takewhile(lambda A__    : x < n    ,						prime_generator()    )    )


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









"""simple docstring"""


def 					lowerCAmelCase				(    __UpperCamelCase ):




  """simple docstring"""







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



def 					lowerCAmelCase				(    __UpperCamelCase ):




  """simple docstring"""







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


if __name__ == "__main__":
       import doctest

       doctest.testmod()
       lowercase_				     =     [
           [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]]))




