




'''simple docstring'''

from __future__ import annotations

from fractions import Fraction




def    _a    (							lowerCAmelCase_			,      lowerCAmelCase_	):





      """simple docstring"""



      return (
          num != den and num % 10 == den // 10 and (num // 10) / (den % 10) == num / den
      )




def    _a    (							lowerCAmelCase_	):





      """simple docstring"""



      _snake_case		:	int					     =     []
      _snake_case		:	Tuple					     =     11
      _snake_case		:	Optional[int]					     =     int('''1''' + '''0''' * digit_len	)
      for num in range(lowerCAmelCase_			,      lowerCAmelCase_	):
            while den <= 99:
                  if (num != den) and (num % 10 == den // 10) and (den % 10 != 0):
                        if is_digit_cancelling(lowerCAmelCase_			,      lowerCAmelCase_	):
                              solutions.append(f'''{num}/{den}'''	)
                  den += 1
            num += 1
            _snake_case		:	int					     =     10
      return solutions




def    _a    (							lowerCAmelCase_ = 2	):





      """simple docstring"""



      _snake_case		:	Optional[int]					     =     1.0
      for fraction in fraction_list(lowerCAmelCase_	):
            _snake_case		:	Optional[Any]					     =     Fraction(lowerCAmelCase_	)
            result *= frac.denominator / frac.numerator
      return int(lowerCAmelCase_	)


if __name__ == "__main__":
   print(solution())









'''simple docstring'''

import math

from numpy import inf
from scipy.integrate import quad




def    _a    (							lowerCAmelCase_	):





      """simple docstring"""



      if num <= 0:
            raise ValueError('''math domain error'''	)

      return quad(lowerCAmelCase_			,      0			,      lowerCAmelCase_			,      args=(lowerCAmelCase_)	)[0]




def    _a    (							lowerCAmelCase_			,      lowerCAmelCase_	):





      """simple docstring"""



      return math.pow(lowerCAmelCase_			,      z - 1	) * math.exp(-x	)


if __name__ == "__main__":
   from doctest import testmod

   testmod()


