Polynomials are in Q[X].

• The polynomial X - 1 is a common divisor of X² - 1 and X³ - 1, since
  X² - 1 = (X - 1) (X + 1) and X³ - 1 = (X - 1)( X² + X + 1).

  An ad hoc proof that (up to a nonzero scalar factor) this is the greatest common divisor runs as follows. Divide X³ - 1 by X² - 1. This yields

  X³ - 1 = X(X² - 1) + X - 1.

  From this identity we read off that every common divisor of X² - 1 and X³ - 1 must also be a divisor of X - 1.

• A common multiple of X² - 1 and X² + 1 is X⁴ - 1. In fact, this is a least common multiple. The proof is easy once you know more about the theory. At this point it would be a tedious computation.