  In this paper, we construct $2n-1$ locally indistinguishable orthogonal
product states in $\mathbb{C}^n\otimes\mathbb{C}^{4}~(n>4)$ and
$\mathbb{C}^n\otimes\mathbb{C}^{5}~(n\geq 5)$ respectively. Moreover, a set of
locally indistinguishable orthogonal product states with $2(n+2l)-8$ elements
in $\mathbb{C}^n\otimes\mathbb{C}^{2l}~(n\geq 2l>4)$ and a class of locally
indistinguishable orthogonal product states with $2(n+2k+1)-7$ elements in
$\mathbb{C}^n\otimes\mathbb{C}^{2k+1}~(n\geq 2k+1>5)$ are also constructed
respectively. These classes of quantum states are then shown to be
distinguishable by local operation and classical communication (LOCC) using a
suitable $\mathbb{C}^2\otimes\mathbb{C}^2$ maximally entangled state
respectively.
