Counting the number of pairs of $(i,j)$ such that $\operatorname{lcm}(i,j)=n$ Pairs $(i,j)$ such that $\operatorname{lcm}(i,j)=n$
how many pair $(i,j)$ can be formed such that $\operatorname{lcm}(i,j)=n$. here  $i\leq n$ and $j\leq n$.
here $n$ is a integer number less than $10^{14}$.

This can get you started.
Factor $n$ into its prime decomposition:
$$n=p_1^{a_1}p_2^{a_2}\dots p_k^{a_k}$$
All the possible $i$'s where $\operatorname{lcm}(i,j)=n$ are given by
$$i=p_1^{b_1}p_2^{b_2}\dots p_k^{b_k}$$
where $0\le b_r\le a_r$. This gives $a_r+1$ choices for the exponent for $p_r$, and the choices are independent for each $r$, so the number of possible choices for $i$ is
$$(a_1+1)(a_2+1)\ldots (a_k+1)$$
Continue from there. The main difficulty is that the number of possible choices for $j$ depends on the particular $i$.
Show us some of the work you do from here, and we can give you more ideas.
