Question 81
An expression $\frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f}$ is written on the blackboard (numbers $a$, $b$, $c$, $d$, $e$, and $f$ are positive integers). If one increases $a$ by $1$, then the value of this expression increases by $3$. If one increases $c$ by $1$ in the initial expression, then its value increases by $4$. If one increases $e$ by $1$ in the initial expression, then its value increases by $5$. Find the minimal possible value of the product $bdf$.
(N. Agakhanov)
(N. Agakhanov)