  In this note we shall give conditions which guarantee that
$\frac{1-q^b}{1-q^a}{n\brack m}\in\mathbb{Z}[q]$ holds. We shall provide a full
characterisation for $\frac{1-q^b}{1-q^a}{ka\brack m}\in\mathbb{Z}[q]$. This
unifies a variety of results already known in literature. We shall prove new
divisibility properties for the binomial coefficients and a new divisibility
result for a certain finite sum involving the roots of the unity.
