Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by $$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$ The numbers $p_t(n)$ can be regarded as enumerating partitions of $n$ into parts that come with $t$ colors. Furthermore, $p_t(n)=\sum_{\lambda\vdash n}\prod_{j\geq1}\binom{k_j+t-1}{t-1}$ where $\lambda=1^{k_1}2^{k_2}\cdots$ and each $k_j\geq0$. Note also that $p_1(n)=p(n)$ is the usual number of (unrestricted) integer partitions of $n$. Ramanujan's famous congruences state $$\begin{cases} p(5n+4)\equiv0\mod 5, \\ p(7n+5)\equiv0\mod 7, \\ p(11n+6)\equiv0\mod 11. \end{cases}$$

In the same spirit, the following appear to be true. Are they? $$\begin{cases} p_t(5n+4)\equiv0\mod 5, \qquad t\equiv0,1,2,4\mod 5 \\ \,p_t(7n+5)\equiv0\mod 7, \qquad \,\,t\equiv0,1,4 \,\,\,\, \mod 7\\ p_t(11n+6)\equiv0\mod 11, \qquad t\equiv0,1,10\mod 11. \end{cases}$$