Ramanujan-type Congruences for Partition $k$-Tuples with $5$-Cores

We prove several Ramanujan-type congruences modulo powers of $5$ for partition $k$-tuples with $5$-cores, for $k=2,3,4$. We also prove some new infinite families of congruences modulo powers of primes for $k$-tuples with $p$-cores, where $p$ is a prime.