






"""simple docstring"""




def 			lowerCAmelCase__     (     UpperCamelCase__ = 1_0_0_0_0_0_0				):



  '''simple docstring'''
  _a  :       Optional[int]             =	set(range(3     ,   UpperCamelCase__     ,   2				)				)
  primes.add(2				)
  for p in range(3     ,   UpperCamelCase__     ,   2				):
    if p not in primes:
      continue
    primes.difference_update(set(range(p * p     ,   UpperCamelCase__     ,   UpperCamelCase__				)				)				)

  _a  :       List[Any]             =	[float(UpperCamelCase__				) for n in range(limit + 1				)]

  for p in primes:
    for n in range(UpperCamelCase__     ,   limit + 1     ,   UpperCamelCase__				):
      phi[n] *= 1 - 1 / p

  return int(sum(phi[2:]				)				)


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









"""simple docstring"""




def 			lowerCAmelCase__     (     UpperCamelCase__     ,   UpperCamelCase__				):



  '''simple docstring'''

  # Check if the input is valid
  if not len(UpperCamelCase__				) == len(UpperCamelCase__				) == 3:
    raise ValueError("""Please enter a valid equation."""				)
  if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
    raise ValueError("""Both a & b of two equations can't be zero."""				)

  # Extract the coefficients
  _a      ,  _a      ,  _a  :       Any             =	equationa
  _a      ,  _a      ,  _a  :       Tuple             =	equationa

  # Calculate the determinants of the matrices
  _a  :       int             =	aa * ba - aa * ba
  _a  :       str             =	ca * ba - ca * ba
  _a  :       str             =	aa * ca - aa * ca

  # Check if the system of linear equations has a solution (using Cramer's rule)
  if determinant == 0:
    if determinant_x == determinant_y == 0:
      raise ValueError("""Infinite solutions. (Consistent system)"""				)
    else:
      raise ValueError("""No solution. (Inconsistent system)"""				)
  else:
    if determinant_x == determinant_y == 0:
      # Trivial solution (Inconsistent system)
      return (0.0, 0.0)
    else:
      _a  :       Dict             =	determinant_x / determinant
      _a  :       str             =	determinant_y / determinant
      # Non-Trivial Solution (Consistent system)
      return (x, y)

