
def 	a_		(	__lowercase       :     int = 1_000_000			)			->  int:
			_snake_case          =      set(range(3	, __lowercase	, 2			)			)
			primes.add(2			)
			for p in range(3	, __lowercase	, 2			):
						if p not in primes:
									continue
						primes.difference_update(set(range(p * p	, __lowercase	, __lowercase			)			)			)

			_snake_case          =      [float(__lowercase			) for n in range(limit + 1			)]

			for p in primes:
						for n in range(__lowercase	, limit + 1	, __lowercase			):
									phi[n] *= 1 - 1 / p

			return int(sum(phi[2:]			)			)


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

def 	a_		(	__lowercase       :     list[int]	, __lowercase       :     list[int]			)			->  tuple[float, float]:

			# Check if the input is valid
			if not len(__lowercase			) == len(__lowercase			) == 3:
						raise ValueError('Please enter a valid equation.'			)
			if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
						raise ValueError('Both a & b of two equations can\'t be zero.'			)

			# Extract the coefficients
			_snake_case     , _snake_case     , _snake_case          =      equationa
			_snake_case     , _snake_case     , _snake_case          =      equationa

			# Calculate the determinants of the matrices
			_snake_case          =      aa * ba - aa * ba
			_snake_case          =      ca * ba - ca * ba
			_snake_case          =      aa * ca - aa * ca

			# Check if the system of linear equations has a solution (using Cramer's rule)
			if determinant == 0:
						if determinant_x == determinant_y == 0:
									raise ValueError('Infinite solutions. (Consistent system)'			)
						else:
									raise ValueError('No solution. (Inconsistent system)'			)
			else:
						if determinant_x == determinant_y == 0:
									# Trivial solution (Inconsistent system)
									return (0.0, 0.0)
						else:
									_snake_case          =      determinant_x / determinant
									_snake_case          =      determinant_y / determinant
									# Non-Trivial Solution (Consistent system)
									return (x, y)