





from __future__ import annotations

from functools import lru_cache
from math import ceil

__lowercase       :Tuple       =  100

__lowercase       :Optional[int]       =  set(range(3, NUM_PRIMES, 2))
primes.add(2)
__lowercase       :int

for prime in range(3, ceil(NUM_PRIMES**0.5), 2):
				if prime not in primes:
								continue
				primes.difference_update(set(range(prime * prime, NUM_PRIMES, prime)))







@lru_cache(maxsize=100     )
def       UpperCAmelCase							(     _lowerCamelCase      :  int     ):





						'''simple docstring'''


						if number_to_partition < 0:
												return set()
						elif number_to_partition == 0:
												return {1}

						SCREAMING_SNAKE_CASE__     :				set[int]   =					set()
						SCREAMING_SNAKE_CASE__     :				int
						SCREAMING_SNAKE_CASE__     :				int

						for prime in primes:
												if prime > number_to_partition:
																		continue
												for sub in partition(number_to_partition - prime     ):
																		ret.add(sub * prime     )

						return ret







def       UpperCAmelCase							(     _lowerCamelCase      :  int = 5_000     ):





						'''simple docstring'''


						for number_to_partition in range(1			,   _lowerCamelCase     ):
												if len(partition(_lowerCamelCase     )     ) > number_unique_partitions:
																		return number_to_partition
						return None


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






from __future__ import annotations

from fractions import Fraction







def       UpperCAmelCase							(     _lowerCamelCase      :  int			,   _lowerCamelCase      :  int     ):





						'''simple docstring'''


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







def       UpperCAmelCase							(     _lowerCamelCase      :  int     ):





						'''simple docstring'''


						SCREAMING_SNAKE_CASE__     :				int   =					[]
						SCREAMING_SNAKE_CASE__     :				str   =					11
						SCREAMING_SNAKE_CASE__     :				Any   =					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
												SCREAMING_SNAKE_CASE__     :				str   =					10
						return solutions







def       UpperCAmelCase							(     _lowerCamelCase      :  int = 2     ):





						'''simple docstring'''


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


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