  Let $\langle A\rangle$ be the numerical semigroup generated by relatively
prime positive integers $\{a_1,a_2,...,a_n\}$. The quotient of $\langle
A\rangle$ with respect to a positive integer $p$ is defined by $\frac{\langle
A\rangle}{p}=\{x\in \mathbb{N} \mid px\in \langle A\rangle\}$. The quotient
$\frac{\langle A\rangle}{p}$ is known to be a semigroup but is hard to study.
When $p$ is a positive divisor of $a_1$, we reduce the computation of the
Ap\'ery set of $\frac{a_1}{p}$ in $\frac{\langle A\rangle}{p}$ to a simple
minimization problem. This allow us to obtain closed formulas of the Frobenius
number of the quotient for some special numerical semigroups. These includes
the cases when $\langle A\rangle$ is the almost arithmetic progressions, the
almost arithmetic progressions with initial gaps, etc. In particular, we
partially solve an open problem proposed by A. Adeniran et al.
