



from __future__ import annotations

from functools import lru_cache
from math import ceil

snake_case    :						Dict  =						1_0_0

snake_case    :						Union[str, Any]  =						set(range(3, NUM_PRIMES, 2))
primes.add(2)
snake_case    :						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=1_0_0       )
def 	snake_case__       (					__lowercase       )		->		set[int]:







						"""simple docstring"""

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

						A__	:							set[int]						    = set()
						A__	:							int
						A__	:							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 	snake_case__       (					__lowercase = 5_0_0_0       )		->		int | None:







						"""simple docstring"""

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


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




def 	snake_case__       (					__lowercase       )		->		int:







						"""simple docstring"""

						if divisor % 5 == 0 or divisor % 2 == 0:
												return 0
						A__	:							Tuple						    = 1
						A__	:							Union[str, Any]						    = 1
						while repunit:
												A__	:							Union[str, Any]						    = (1_0 * repunit + 1) % divisor
												repunit_index += 1
						return repunit_index
def 	snake_case__       (					__lowercase = 1_0_0_0_0_0_0       )		->		int:







						"""simple docstring"""

						A__	:							Dict						    = limit - 1
						if divisor % 2 == 0:
												divisor += 1
						while least_divisible_repunit(__lowercase       ) <= limit:
												divisor += 2
						return divisor


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