Consider the number of integer partitions $p(n)$ of $n$ whose (product) generating function reads $$\sum_{n\geq0}p(n)\,x^n=\prod_{k\geq1}\frac1{1-x^k}.$$ There are many congruences for $p(n)$ including those due to Ramanujan: $p(5n+4)\equiv_50, p(7n+5)\equiv_70$ and $p(11n+6)\equiv_{11}0$. Here, I would like to ask:
QUESTION. Is this true? For any prime $q$, $$\#\{p(n)\,\,\, \text{mod}\,\,\, q: n\in\mathbb{N}\}=q.$$ That means, do all modular residues appear? In particular, given a prime $q$, does there always exist an $n$ such that $p(n)\,\,\,\text{mod}\,\,\, q=0$?