  A remarkable theorem of Joris states that a function $f$ is $C^\infty$ if two
relatively prime powers of $f$ are $C^\infty$. Recently, Thilliez showed that
an analogous theorem holds in Denjoy--Carleman classes of Roumieu type. We
prove that a division property, equivalent to Joris's result, is valid in a
wide variety of ultradifferentiable classes. Generally speaking, it holds in
all dimensions for non-quasianalytic classes. In the quasianalytic case we have
general validity in dimension one, but we also get validity in all dimensions
for certain quasianalytic classes.
