"""simple docstring"""


from __future__ import annotations

from fractions import Fraction







def 					__a   (		A    ,				A      ):






     '''simple docstring'''



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







def 					__a   (		A      ):






     '''simple docstring'''



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







def 					__a   (		A = 2      ):






     '''simple docstring'''



     lowercase__				 =							1.0
     for fraction in fraction_list(A      ):
          lowercase__				 =							Fraction(A      )
          result *= frac.denominator / frac.numerator
     return int(A      )


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





"""simple docstring"""


from sympy import diff, lambdify, symbols
from sympy.functions import *  # noqa: F403







def 					__a   (		A    ,				A    ,				A = "x"    ,				A = 10**-10    ,				A = 1    ,				):






     '''simple docstring'''



     lowercase__				 =							symbols(A      )
     lowercase__				 =							lambdify(A    ,				A      )
     lowercase__				 =							lambdify(A    ,				diff(A    ,				A      )      )

     lowercase__				 =							starting_point

     while True:
          if diff_function(A      ) != 0:
               lowercase__				 =							prev_guess - multiplicity * func(A      ) / diff_function(
                   A      )
          else:
               raise ZeroDivisionError("Could not find root"      ) from None

          # Precision is checked by comparing the difference of consecutive guesses
          if abs(next_guess - prev_guess      ) < precision:
               return next_guess

          lowercase__				 =							next_guess


# Let's Execute
if __name__ == "__main__":
 # Find root of trigonometric function
 # Find value of pi
 print(F'The root of sin(x) = 0 is {newton_raphson("sin(x)", 2)}')

 # Find root of polynomial
 # Find fourth Root of 5
 print(F'The root of x**4 - 5 = 0 is {newton_raphson("x**4 -5", 0.4 +5j)}')

 # Find value of e
 print(
     "The root of log(y) - 1 = 0 is ",
     F'{newton_raphson("log(y) - 1", 2, variable="y")}',
 )

 # Exponential Roots
 print(
     "The root of exp(x) - 1 = 0 is",
     F'{newton_raphson("exp(x) - 1", 1_0, precision=0.005)}',
 )

 # Find root of cos(x)
 print(F'The root of cos(x) = 0 is {newton_raphson("cos(x)", 0)}')




