





"""simple docstring"""

from __future__ import annotations

from fractions import Fraction






def 							A						(    snake_case				:int    , snake_case				:int		)				->     bool:
 return (
     num != den and num % 1_0 == den // 1_0 and (num // 1_0) / (den % 1_0) == num / den
 )






def 							A						(    snake_case				:int		)				->     list[str]:
 __UpperCamelCase      =	[]
 __UpperCamelCase      =	1_1
 __UpperCamelCase      =	int('1' + '0' * digit_len		)
 for num in range(snake_case    , snake_case		):
  while den <= 9_9:
   if (num != den) and (num % 1_0 == den // 1_0) and (den % 1_0 != 0):
    if is_digit_cancelling(snake_case    , snake_case		):
     solutions.append(f'{num}/{den}'		)
   den += 1
  num += 1
  __UpperCamelCase      =	1_0
 return solutions






def 							A						(    snake_case				:int = 2		)				->     int:
 __UpperCamelCase      =	1.0
 for fraction in fraction_list(snake_case		):
  __UpperCamelCase      =	Fraction(snake_case		)
  result *= frac.denominator / frac.numerator
 return int(snake_case		)


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





"""simple docstring"""

def 							A						(    snake_case				:int		)				->     int:
 __UpperCamelCase      =	abs(snake_case		)
 __UpperCamelCase      =	0
 while n > 0:
  res += n % 1_0
  n //= 1_0
 return res






def 							A						(    snake_case				:int		)				->     int:
 __UpperCamelCase      =	abs(snake_case		)
 return n if n < 1_0 else n % 1_0 + sum_of_digits(n // 1_0		)






def 							A						(    snake_case				:int		)				->     int:
 return sum(int(snake_case		) for c in str(abs(snake_case		)		)		)






def 							A						(    )				->     None:
 from collections.abc import Callable
 from timeit import timeit

 def benchmark_a_function(snake_case				:Callable    , snake_case				:int		) -> None:
  __UpperCamelCase      =	f'{func.__name__}({value})'
  __UpperCamelCase      =	timeit(f'__main__.{call}'    , setup='import __main__'		)
  print(f'{call:56} = {func(snake_case		)} -- {timing:.4f} seconds'		)

 for value in (2_6_2_1_4_4, 1_1_2_5_8_9_9_9_0_6_8_4_2_6_2_4, 1_2_6_7_6_5_0_6_0_0_2_2_8_2_2_9_4_0_1_4_9_6_7_0_3_2_0_5_3_7_6):
  for func in (sum_of_digits, sum_of_digits_recursion, sum_of_digits_compact):
   benchmark_a_function(snake_case    , snake_case		)
  print()


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      benchmark()



