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

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