Ltac zero_or_not a :=
  destruct (Z.eq_dec a 0);
  [subst; rewrite ?Zmod_0_l, ?Zdiv_0_l, ?Zmod_0_r, ?Zdiv_0_r;
   auto with zarith|].

Lemma Z_div_exact_full_2 : forall a b:Z, b <> 0 -> a mod b = 0 -> a = b*(a/b).