  Let $N$ be a positive integer and let $J_0(N)$ be the Jacobian variety of the
modular curve $X_0(N)$. For any prime $p\ge 5$ whose square does not divide
$N$, we prove that the $p$-primary subgroup of the rational torsion subgroup of
$J_0(N)$ is equal to that of the rational cuspidal divisor class group of
$X_0(N)$, which is explicitly computed in \cite{Yoo9}. Also, we prove the same
assertion holds for $p=3$ under the extra assumption that either $N$ is not
divisible by $3$ or there is a prime divisor of $N$ congruent to $-1$ modulo
$3$.
