Let A be a gentle algebra, it is proved that different -rigid A-modules have different dimension vectors if and only if A has no even oriented cycle with full relations. As an application, we establish the denominator conjecture, which was proposed by Fomin and Zelevinsky in 2004, for cluster algebras of type ABC. Namely, for cluster algebras of type ABC, we show that different cluster monomials have different dimension vectors with respect to any initial seeds. This is a joint work with Changjian Fu.