KöMaL 2009 A483

For abritrary integers 0<$k \le n$ and $a$>1, define ${\binom{n}{k}}_{a} = \frac{(a^n-1)(a^{n-1}-1)\cdot \ldots \cdot(a^{n-k+1}-1)}{(a^k-1)(a^{k-1}-1) \cdot \ldots \cdot(a-1)}$. ($a$) Show that ${\binom{n}{k}}_{a}$ is an integer. ($b$) Do there exist integers 0<$k$<$n$<$m$and $a$>1, for which ${\binom{m}{1}}_{a}$ divides ${\binom{n}{k}}_{a}$?