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?
>  $$\begin{cases} p\_t(5n+4)\equiv0\mod 5, \qquad t\equiv0,1,2,4\mod 5 \\ \,p\_t(7n+5)\equiv0\mod 7, \qquad \,\,t\equiv0,1,4 \,\,\,\, \mod 7\\ 
> p\_t(11n+6)\equiv0\mod 11, \qquad t\equiv0,1,10\mod 11. \end{cases}$$
> 
> 
> 



Congruences Ramanujan-style
More general versions of this have been established: see in particular Theorem 2 of [Kiming and Olsson](https://link.springer.com/content/pdf/10.1007/BF01197051.pdf), and for other work see (for example) [Locus and Wagner](http://www.mathcs.emory.edu/~mlocus/CongruencePPF.pdf). 


To answer the question fully, as Ofir Gorodetsky observed the problem is trivial when $t$ is $0$ or $1$ modulo the prime modulus. The other cases correspond to $t \equiv \ell-1$ or $t\equiv \ell-3$ modulo $\ell$ (with $\ell$ being $5$, $7$, or $11$). These cases are the "non-exceptional congruences" covered by Theorem 2 of the paper by Kiming and Olsson (and the argument there is essentially elementary following easily from Euler's pentagonal number theorem in the case of $\ell-1$, and an identity of Jacobi in the case of $\ell-3$). The real interest is in the situation of exceptional congruences, and Theorem 4 of Kiming and Olsson gives some examples of these. For instance, if $t\equiv 3 \mod 11$ then 
$$ 
p\_{t}(11n+7) \equiv 0 \pmod{11}. 
$$ 


