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_mod_mult : forall a b, (a*b) mod b = 0.