I am looking for a reference for a proof of the following:
Let $n$ and $a,b, \ldots ,z$ be non-negative integers with $a + b + \ldots + z = n$, and let $p$ be a prime. Write $n = n_0 + n_1 p + \ldots + n_m p^m$ in $p$-ary notation, similarly for $a, b, \ldots , z$. Then, modulo $p$, the multinomial coefficient ${n \choose a,b, \ldots, z}$ is zero if there is some "carrying" in computing the sum $a + b + \ldots + z$ (i.e., $a_i + b_i + \ldots + z_i \geq p$ for some $i$). Otherwise, it is the product of the multinomial coefficients of the individual $p$-digits, that is,
${ n \choose a, b, \ldots, z} = {n_0 \choose a_0, b_0, \ldots, z_0}{n_1 \choose a_1, b_1, \ldots, z_1} \ldots {n_m \choose a_m, b_m, \ldots , z_m}$
I've found plenty of references for this theorem restricted to binomial coefficients, otherwise known as Lucas' theorem, but can't seem to find one for arbitrary multinomials.
Thanks in advance for the help.