Infinite families of pairs of real quadratic fields whose class numbers are divisible by a given integer

For any integers $N\ge 2$ and $m\ge 1$, we prove that there exist infinitely many pairs of real quadratic fields $\text{QQ}(\sqrt{D}),\text{QQ}(\sqrt{D+m})$, with $D\in \text{ZZ}$ and $D>0$, such that the class numbers of both fields are divisible by $N$. Write $N=2^{e}n$ with $n$ odd. If $n>1$, then the class groups of both fields in each such pair contain an element of order $n$.