There are $20$ horses, named Horse 1, Horse 2, $\dots$, Horse 20. Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S>0$, in minutes, at which all $20$ horses will again simultaneously be at the starting point is $S=\text{LCM}(1,2,\dots ,20)$. Let $T>0$ be the least time, in minutes, such that at least $8$ of the horses are again at the starting point. What is the sum of the digits of $T$?