Let $k = Q(sqrt{m})$ be a real quadratic field. In this paper, the following theorems on p-divisibility of the class number h of k are studied for each prime pp. Theorem 1. If the discriminant of k has at least three distinct prime divisors, then 2 divides h. Theorem 2. If an odd prime p divides h, then p divides $B_{a,chiomega^{-1}}$, where $chi$ is the nontrivial character of k, and $omega$ is the Teichmuller character for pp. Theorem 3. Let $h_n$ be the class number of $k_n$, the nth layer of the $Z_p$-extension $k_infty$ of k. If p does not divide $B_{a,chiomega^{-1}}$, then $p otmid h_n$ for all $n geq 0$.