






from __future__ import annotations

from fractions import Fraction

def        UpperCAmelCase__		(						lowerCamelCase_       :       int   ,  lowerCamelCase_       :       int				):
			return (
			    num != den and num % 1_0 == den // 1_0 and (num // 1_0) / (den % 1_0) == num / den
			)

def        UpperCAmelCase__		(						lowerCamelCase_       :       int				):
			__a						:  int									=     []
			__a						:  int									=     1_1
			__a						:  List[Any]									=     int('1' + '0' * digit_len				)
			for num in range(lowerCamelCase_   ,  lowerCamelCase_				):
						while den <= 9_9:
									if (num != den) and (num % 1_0 == den // 1_0) and (den % 1_0 != 0):
												if is_digit_cancelling(lowerCamelCase_   ,  lowerCamelCase_				):
															solutions.append(f'''{num}/{den}'''				)
									den += 1
						num += 1
						__a						:  Union[str, Any]									=     1_0
			return solutions

def        UpperCAmelCase__		(						lowerCamelCase_       :       int = 2				):
			__a						:  Optional[Any]									=     1.0
			for fraction in fraction_list(lowerCamelCase_				):
						__a						:  int									=     Fraction(lowerCamelCase_				)
						result *= frac.denominator / frac.numerator
			return int(lowerCamelCase_				)


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













import unittest

import numpy as np





def 		__UpperCAmelCase     (	a_				,	a_				,	a_				,	a_ = None				,	):
							snake_case_           =				np.shape(a_)
							snake_case_           =				np.shape(a_)
							snake_case_           =				np.shape(a_)

							if shape_a[0] != shape_b[0]:
														snake_case_           =				(
														    'Expected the same number of rows for A and B. '
														    f'''Instead found A of size {shape_a} and B of size {shape_b}'''
														)
														raise ValueError(a_)

							if shape_b[1] != shape_c[1]:
														snake_case_           =				(
														    'Expected the same number of columns for B and C. '
														    f'''Instead found B of size {shape_b} and C of size {shape_c}'''
														)
														raise ValueError(a_)

							snake_case_           =				pseudo_inv
							if a_inv is None:
														try:
																					snake_case_           =				np.linalg.inv(a_)
														except np.linalg.LinAlgError:
																					raise ValueError(
																					    'Input matrix A is not invertible. Cannot compute Schur complement.')

							return mat_c - mat_b.T @ a_inv @ mat_b




class 					UpperCamelCase_ (     unittest.TestCase							):



	'''simple docstring'''




	def 				_UpperCamelCase     (			self   )			->       None:
								snake_case_           =				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]   )
								snake_case_           =				np.array([[0, 3], [3, 0], [2, 3]]   )
								snake_case_           =				np.array([[2, 1], [6, 3]]   )

								snake_case_           =				schur_complement(a			,     a			,     a   )

								snake_case_           =				np.block([[a, b], [b.T, c]]   )

								snake_case_           =				np.linalg.det(a   )
								snake_case_           =				np.linalg.det(a   )
								snake_case_           =				np.linalg.det(a   )

								self.assertAlmostEqual(a			,     det_a * det_s   )




	def 				_UpperCamelCase     (			self   )			->       None:
								snake_case_           =				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]   )
								snake_case_           =				np.array([[0, 3], [3, 0], [2, 3]]   )
								snake_case_           =				np.array([[2, 1], [6, 3]]   )

								with self.assertRaises(a   ):
															schur_complement(a			,     a			,     a   )






	def 				_UpperCamelCase     (			self   )			->       None:
								snake_case_           =				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]   )
								snake_case_           =				np.array([[0, 3], [3, 0], [2, 3]]   )
								snake_case_           =				np.array([[2, 1, 3], [6, 3, 5]]   )

								with self.assertRaises(a   ):
															schur_complement(a			,     a			,     a   )


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	unittest.main()





