


from __future__ import annotations

from fractions import Fraction







def        _lowerCamelCase(	lowercase__   , lowercase__	)						->	bool:

		'''simple docstring'''







		return (
		    num != den and num % 1_0 == den // 1_0 and (num // 1_0) / (den % 1_0) == num / den
		)







def        _lowerCamelCase(	lowercase__	)						->	list[str]:

		'''simple docstring'''







		__lowercase=			[]
		__lowercase=			1_1
		__lowercase=			int('1' + '0' * digit_len	)
		for num in range(lowercase__   , lowercase__	):
				while den <= 9_9:
						if (num != den) and (num % 1_0 == den // 1_0) and (den % 1_0 != 0):
								if is_digit_cancelling(lowercase__   , lowercase__	):
										solutions.append(F'{num}/{den}'	)
						den += 1
				num += 1
				__lowercase=			1_0
		return solutions







def        _lowerCamelCase(	lowercase__ = 2	)						->	int:

		'''simple docstring'''







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


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










from math import isqrt







def        _lowerCamelCase(	lowercase__	)						->	bool:

		'''simple docstring'''







		return all(number % divisor != 0 for divisor in range(2   , isqrt(lowercase__	) + 1	)	)







def        _lowerCamelCase(	lowercase__ = 1_0**6	)						->	int:

		'''simple docstring'''







		__lowercase=			0
		__lowercase=			1
		__lowercase=			7
		while prime_candidate < max_prime:
				primes_count += is_prime(lowercase__	)

				cube_index += 1
				prime_candidate += 6 * cube_index

		return primes_count


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






