Show that number of triples $(a,b,c)$ with $a,b,c\in [1,n]$ such that $ab=c$ is given by
$$\bigl|\bigl\{(a,b,c)\in [1,n]^3:ab=c\bigr\}\bigr|=2\sum_{i=1}^{\left\lfloor\sqrt{n}\right\rfloor}\Big(\left\lfloor\frac ni\right\rfloor-i\Big)+\left\lfloor\sqrt{n}\right\rfloor$$
Someone told me that this can be solved using hyperbolas. I could not really understand how that would help. Please help!
It could have been easier if the interval were not a requirement, but that is the part that is confusing me the most.
from Hot Weekly Questions - Mathematics Stack Exchange
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