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