  Let $NT(m, k, n)$ denote the total number of parts in the partitions of n
with rank congruent to m modulo k. Andrews proved Beck's conjecture on
congruences for $NT(m, k, n)$ modulo 5 and 7. Generalizing Andrews'results,
Chern obtain congruences for $NT(m, k, n)$ modulo 11 and 13. More recently, the
second author use the theory of Hecke operators to establish congruences for
such partition statistics modulo powers of primes $\ell \ge 7$. In this paper,
we obtain Andrews-Beck type congruences modulo powers of 5.
