Let $p$ and $q$ be distinct primes such that $p \equiv q \pmod{4}$ and consider the quadratic field $K = \mathbf{Q}(\sqrt{pq})$. In this paper, we shall investigate the class group and determine the exact power of 2 dividing the class number of $K$ using the theory of ideals and a theorem on the solvability of $ax^2 + by^2 = z^2$.