'''simple docstring'''




def _lowercase						(   lowerCamelCase__ = 100_0000    )       ->							Union[str, Any]:





       """simple docstring"""


       __UpperCAmelCase	:					str												=		set(range(3				,     __snake_case				,     2    )    )
       primes.add(2    )
       for p in range(3				,     __snake_case				,     2    ):
              if p not in primes:
                     continue
              primes.difference_update(set(range(p * p				,     __snake_case				,     __snake_case    )    )    )

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

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

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


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










"""simple docstring"""
from __future__ import annotations

from numpy import array, cos, cross, floataa, radians, sin
from numpy.typing import NDArray

def 				_snake_case      (   __snake_case							:      float						, __snake_case							:      float						, __snake_case							:      bool = False   ):




   """simple docstring"""







   if radian_mode:
      return [magnitude * cos(__snake_case   ), magnitude * sin(__snake_case   )]
   return [magnitude * cos(radians(__snake_case   )   ), magnitude * sin(radians(__snake_case   )   )]

def 				_snake_case      (   __snake_case							:      NDArray[floataa]						, __snake_case							:      NDArray[floataa]						, __snake_case							:      float = 10**-1   ):




   """simple docstring"""







   _lowerCamelCase	:					NDArray[floataa]   =		cross(__snake_case						, __snake_case   )
   _lowerCamelCase	:					float   =		sum(__snake_case   )
   return abs(__snake_case   ) < eps


if __name__ == "__main__":
 # Test to check if it works
 UpperCAmelCase				=						array(
     [
         polar_force(718.4, 180 - 30),
         polar_force(879.54, 45),
         polar_force(100, -90),
     ]
 )

 UpperCAmelCase				=						array([[0, 0], [0, 0], [0, 0]])

 assert in_static_equilibrium(forces, location)

 # Problem 1 in image_data/2D_problems.jpg
 UpperCAmelCase				=						array(
     [
         polar_force(30 * 9.81, 15),
         polar_force(215, 180 - 45),
         polar_force(264, 90 - 30),
     ]
 )

 UpperCAmelCase				=						array([[0, 0], [0, 0], [0, 0]])

 assert in_static_equilibrium(forces, location)

 # Problem in image_data/2D_problems_1.jpg
 UpperCAmelCase				=						array([[0, -2000], [0, -1200], [0, 1_5600], [0, -1_2400]])

 UpperCAmelCase				=						array([[0, 0], [6, 0], [10, 0], [12, 0]])

 assert in_static_equilibrium(forces, location)

 import doctest

 doctest.testmod()

