A way to solve equations of the form $(\frac{n}{a})! = b$?

Question

I found myself today needing the natural solutions for $n$ satisfying this equation:
$(\frac{n}{5})! = 2$

In this case I "got lucky" and was able to guess that 10 is a suitable solution.

Here are questions that popped in my mind:

1. Probably a trivial answer, but is this the only solution? If yes, how can we show it formally?
2. Perhaps a more interesting question:
   given two constants $a,b \in \Bbb{N}$ , is there a way for calculating all Natural solutions for $n$ satisfying: $(\frac{n}{a})!=b$
   ?

Answer 1

$(n/a)!$ is (rapidly) increasing. So there can be at most one solution of $(n/a)!=b$ which can be easily found by checking $n=a, 2a,...$.