Arithmetic properties of $(\mathrm{\ell },m)$-regular colored partitions

Let $b_{\mathrm{\ell },m}^k(n)$ denotes the number of $k-$colored partitions of $n$ into parts that are not multiples of $\mathrm{\ell }$ or $m$. We establish several congruence relations for $b_{\mathrm{\ell },m}(n)$. For instance, for any nonnegative integer $n$ $$b_{4,5}^2(8n+7)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}40)\mathrm{.}$$