among the numbers between $1100$ and $1159$, every number $n$ less than or equal to $60$ divides at least one of the numbers considered, and therefore does not work. Similarly, given $60 \leq n \leq 80$, $n$ divides exactly two numbers between $1100$ and $1259$ but at most one of them can fall among the numbers that do not appear on the clock (that is, the numbers from $1160$ to $119$) because these are $40$: hence $n$ certainly divides one of the numbers that appear on the clock. We have thus ruled out all $n \leq 80$. On the other hand, a direct computation gives us that $81 \times 15=1215$, $82 \times 15=1230$, $83 \times 15=1245$ and therefore these $n$ are not acceptable, while we have $84 \times 13=1092<1100$, $84 \times 14=1176>1159$ and $84 \times 15=1260>1259$.