It was proved here that if $a\in \mathbb{N}_{\geq3}$ then

$$\bigcap_{i = 1}^{a} \bigcup_{j = 0}^{i-1} \left[\frac{1+aj}{i},\frac{a(j+1)-1}{i}\right] = \varnothing \tag{1}$$

It may be conjectured that forcing $i\ne b$, where $1< b< a$, renders $(1)$ untrue, that is, the result is not an empty interval.

I tried analyzing the gaps but everything seems to meet up at a dead end.

What tools may one employ to handle such problems?