  For $G=G_{n, 1/2}$, the Erd\H{o}s--Renyi random graph, let $X_n$ be the
random variable representing the number of distinct partitions of $V(G)$ into
sets $A_1, \ldots, A_q$ so that the degree of each vertex in $G[A_i]$ is
divisible by $q$ for all $i\in[q]$. We prove that if $q\geq 3$ is odd then
$X_n\xrightarrow{d}{\mathrm{Po}(1/q!)}$, and if $q \geq 4$ is even then
$X_n\xrightarrow{d}{\mathrm{Po}(2^q/q!)}$. More generally, we show that the
distribution is still asymptotically Poisson when we require all degrees in
$G[A_i]$ to be congruent to $x_i$ modulo $q$ for each $i\in[q]$, where the
residues $x_i$ may be chosen freely. For $q=2$, the distribution is not
asymptotically Poisson, but it can be determined explicitly.
