  Let $p$ be a prime greater than 3. Consider the modular curve $X_0(3p)$ over
$\mathbb{Q}$ and its Jacobian variety $J_0(3p)$ over $\mathbb{Q}$. Let
$\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ be the group of rational torsion points
on $J_0(3p)$ and the cuspidal group of $J_0(3p)$, respectively. We prove that
the $3$-primary subgroups of $\mathcal{T}(3p)$ and $\mathcal{C}(3p)$ coincide
unless $p\equiv 1 \pmod 9$ and $3^{\frac{p-1}{3}} \equiv 1 \!\pmod {p}$.
