Number theory: find $a, b$ such that $\frac{a}{b} = b.a$ in a general base $\mathcal{B}\neq 10$ I was playing with numbers and thinking about this "coincidence"
$$\frac{5}{2} = 2.5$$
That is, for positive $a$ and $b$ we have
$$\frac{a}{b} = b.a$$
And those questions came into my mind:
1. Could we find all such integers pair $a, b$? (clearly in base $10$)
And due to the fact that we work in base $10$, a more general problem popped up, that is, to write our numbers in a base $\mathcal{B}\neq 10$  and thence look for triplets $(\mathcal{B}, a, b)$ such that 
$$\frac{a}{b} = b.a$$
When $a, b$ are written in base $\mathcal{B}$.
2. Could we find a general formula that will produce infinitely many such integer triples? 
I am not really into number theory, except for few little questions, so this is more a sort of "I am asking to you experts in the field" question.
If for some reason this problem is unclear or wrong or impossible, just tell me!
Thank you!

My idea would be to do $\frac{a}{b}=b+\frac{a}{10}$ and this leads to $a=\frac{10b^2}{(10-b)}$. I think this may be a possible solution for case 1. Actually I do not have an idea for $\mathcal{B}\neq 10$.
