  In this article, we first investigate the partitions whose parts are
congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition
classes with modulus $k$ introduced by Andrews. Then, we introduce the
$(k,r)$-overpartitions in which only parts equivalent to $r$ modulo $k$ may be
overlined and we will show that the number of $(k,k)$-overpartitions of $n$
equals the number of partitions of $n$ such that the $k$-th occurrence of a
part may be overlined. Finally, we extend separable integer partition classes
with modulus $k$ to overpartitions and then give the generating function for
$(k,r)$-modulo overpartitions, which are the $(k,r)$-overpartitions satisfying
certain congruence conditions.
