Školjka

%V0 Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair $(a, b)$ of positive integer is called [i]good[/i] if $$ a \lceil b \nu \rceil - b \lfloor a \nu \rfloor = m \text{.} $$ A good pair $(a, b)$ is called [i]excellent[/i] if neither of the pairs $(a - b, b)$ and $(a, b - a)$ is good. (As usual, by $\lfloor x \rfloor$ and $\lceil x \rceil$ we denote the integer numbers such that $x - 1 < \lfloor x \rfloor \leq x$ and $x \leq \lceil x \rceil < x + 1$.) Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.