  In this paper we study $b_5(n)$, the $5$-regular partitions of $n$. Using the
theory of modular forms, we prove several theorems on the divisibility and
distribution properties of $b_5(n)$ modulo prime $m\geq5$. In particular, we
prove that there are infinitely many Ramanujan-type congruences modulo prime
$m\geq5$.
