  For a positive integer $N$, let $\mathscr{C}_N(\mathbb{Q})$ be the rational
cuspidal subgroup of $J_0(N)$ and $\mathscr{C}(N)$ be the rational cuspidal
divisor class group of $X_0(N)$, which are both subgroups of the rational
torsion subgroup of $J_0(N)$. We prove that two groups
$\mathscr{C}_N(\mathbb{Q})$ and $\mathscr{C}(N)$ are equal when $N=p^2M$ for
any prime $p$ and any squarefree integer $M$. To achieve this we show that all
modular units on $X_0(N)$ can be written as products of certain functions
$F_{m, h}$, which are constructed from generalized Dedekind eta functions.
Also, we determine the necessary and sufficient conditions for such products to
be modular units on $X_0(N)$ under a mild assumption.
