3 and 4 are in the set $\{1, 2, \dotsc , 11\}$, which you took the LCM of, so 12 (3 $\times$ 4) is also a divisor of 27720.
We can take advantage of this knowledge and instead of constructing the arithmetic sequence $\frac{1}{27720}, \frac{2}{27720}, \dotsc , \frac{11}{27720}$, we can construct the arithmetic sequence $\frac{2}{27720}, \frac{3}{27720}, \dotsc , \frac{12}{27720}$.
$\frac{12}{27720} = \frac{1}{2310}$, and 2310 $<$ 2520. So 2310 is the correct answer.
Trying this with 13 instead of 12 would require taking the LCM of an entirely different set, and trying it with things like 14 (2 $\times$ 7), which are also divisors of 27720, would still require taking the LCM with 13 involved, because you can't form an arithmetic sequence like this by going from $\frac{12}{27720}$ to $\frac{14}{27720}$ (arithmetic sequences require the differences between successive terms to be constant).