
"""simple docstring"""





def 	__a     (		__lowerCamelCase,		__lowerCamelCase  ):

   # Check if the input is valid
   if not len(__lowerCamelCase  ) == len(__lowerCamelCase  ) == 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
   UpperCAmelCase_     ,   UpperCAmelCase_     ,   UpperCAmelCase_					:					str			     =    equationa
   UpperCAmelCase_     ,   UpperCAmelCase_     ,   UpperCAmelCase_					:					Tuple			     =    equationa

   # Calculate the determinants of the matrices
   UpperCAmelCase_					:					Any			     =    aa * ba - aa * ba
   UpperCAmelCase_					:					Optional[int]			     =    ca * ba - ca * ba
   UpperCAmelCase_					:					int			     =    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:
         UpperCAmelCase_					:					List[str]			     =    determinant_x / determinant
         UpperCAmelCase_					:					Optional[int]			     =    determinant_y / determinant
         # Non-Trivial Solution (Consistent system)
         return (x, y)




"""simple docstring"""





def 	__a     (		__lowerCamelCase  ):
   UpperCAmelCase_					:					List[str]			     =    int(__lowerCamelCase  )
   if n_element < 1:
      UpperCAmelCase_					:					List[Any]			     =    ValueError("a should be a positive number"  )
      raise my_error

   UpperCAmelCase_					:					List[Any]			     =    [1]
   UpperCAmelCase_     ,   UpperCAmelCase_     ,   UpperCAmelCase_					:					Optional[int]			     =    (0, 0, 0)
   UpperCAmelCase_					:					Dict			     =    1
   while index < n_element:
      while hamming_list[i] * 2 <= hamming_list[-1]:
         i += 1
      while hamming_list[j] * 3 <= hamming_list[-1]:
         j += 1
      while hamming_list[k] * 5 <= hamming_list[-1]:
         k += 1
      hamming_list.append(
          min(hamming_list[i] * 2,		hamming_list[j] * 3,		hamming_list[k] * 5  )  )
      index += 1
   return hamming_list


if __name__ == "__main__":
    _a		       =							input('Enter the last number (nth term) of the Hamming Number Series: ')
    print('Formula of Hamming Number Series => 2^i * 3^j * 5^k')
    _a		       =							hamming(int(n))
    print('-----------------------------------------------------')
    print(f"""The list with nth numbers is: {hamming_numbers}""")
    print('-----------------------------------------------------')


