Notation Zgcd_inv_0_l := Z.gcd_eq_0_l (only parsing).Notation Zgcd_inv_0_r := Z.gcd_eq_0_r (only parsing).

Theorem Zgcd_div_swap0 : forall a b : Z, 0 < Z.gcd a b -> 0 < b -> (a / Z.gcd a b) * b = a * (b/Z.gcd a b).