




'''simple docstring'''





def 					UpperCamelCase_(     snake_case							:	list[int]							,    snake_case							:	list[int]					):

							'''simple docstring'''






							if not len(snake_case					) == len(snake_case					) == 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)










'''simple docstring'''





def 					UpperCamelCase_(     snake_case							:	int = 1_0_0_0_0_0_0					):

							'''simple docstring'''






							snake_case_      =				set(range(3							,    snake_case							,    2					)					)
							primes.add(2					)
							for p in range(3							,    snake_case							,    2					):
														if p not in primes:
																					continue
														primes.difference_update(set(range(p * p							,    snake_case							,    snake_case					)					)					)

							snake_case_      =				[float(snake_case					) for n in range(limit + 1					)]

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

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


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




