Number of Factors of an Integer

For any interval $\mathcal{A}$ in the real number line not containing zero, define its reciprocal to be the set of numbers of the form $\frac 1x$ where $x$ is an element in $\mathcal{A}$. Compute the number of ordered pairs of positive integers $(m,n)$ with $m< n$ such that the length of the interval $[m,n]$ is $10^{10}$ times the length of its reciprocal.

(Source: NIMO 16)