


def        snake_case_					(  snake_case = 3      ,							snake_case = 7      ,							snake_case = 1_00_00_00					)  ->    int:
       lowercase__:       Optional[Any]        =       0
       lowercase__:       int        =       1

       for current_denominator in range(1      ,							limit + 1					):
              lowercase__:       Dict        =       current_denominator * numerator // denominator
              if current_denominator % denominator == 0:
                     current_numerator -= 1
              if current_numerator * max_denominator > current_denominator * max_numerator:
                     lowercase__:       str        =       current_numerator
                     lowercase__:       Dict        =       current_denominator
       return max_numerator


if __name__ == "__main__":
 print(solution(numerator=3, denominator=7, limit=1_00_00_00))



from math import pow




def        snake_case_					(  snake_case      ,							snake_case      ,							snake_case      ,							snake_case      ,							snake_case      ,							)  ->    tuple[int, int]:
       if current_sum == needed_sum:
              # If the sum of the powers is equal to needed_sum, then we have a solution.
              solutions_count += 1
              return current_sum, solutions_count

       lowercase__:       int        =       int(pow(snake_case      ,							snake_case					)					)
       if current_sum + i_to_n <= needed_sum:
              # If the sum of the powers is less than needed_sum, then continue adding powers.
              current_sum += i_to_n
              lowercase__      ,						lowercase__:       Any        =       backtrack(
                  snake_case      ,							snake_case      ,							current_number + 1      ,							snake_case      ,							snake_case					)
              current_sum -= i_to_n
       if i_to_n < needed_sum:
              # If the power of i is less than needed_sum, then try with the next power.
              lowercase__      ,						lowercase__:       Optional[int]        =       backtrack(
                  snake_case      ,							snake_case      ,							current_number + 1      ,							snake_case      ,							snake_case					)
       return current_sum, solutions_count




def        snake_case_					(  snake_case      ,							snake_case					)  ->    int:
       if not (1 <= needed_sum <= 10_00 and 2 <= power <= 10):
              raise ValueError(
                  'Invalid input\n'
                  'needed_sum must be between 1 and 1000, power between 2 and 10.'					)

       return backtrack(snake_case      ,							snake_case      ,							1      ,							0      ,							0					)[1]  # Return the solutions_count


if __name__ == "__main__":
 import doctest

 doctest.testmod()


