  We establish a supercongruence conjectured by Almkvist and Zudilin, by
proving a corresponding $q$-supercongruence. Similar $q$-supercongruences are
established for binomial coefficients and the Ap\'{e}ry numbers, by means of a
general criterion involving higher derivatives at roots of unity. Our methods
lead us to discover new examples of the cyclic sieving phenomenon, involving
the $q$-Lucas numbers.
