You are given three ropes and a lighter. The time required to burn through rope $1$ with a single flame is $20$ minutes. The time required to burn through rope $2$ with a single flame is $30$ minutes. The time required to burn through rope $3$ with a single flame is $60$ minutes. The burn rates in different sections of the rope are unknown and vary. For how many integers $m$ in the set $\{5, 10, 15, \ldots, 100, 105, 110\}$ is it NOT possible to measure out exactly $m$ minutes using the three ropes and the lighter?
I know that $20, 30, 50, 60, 80, 90, 110$ are all possible by just lighting the ropes after one finishes after the other (to get $110$, for example, light the $20$ minute rope, when it finishes, light the $30$ minute rope, when it finishes light the $60$ minute rope). However, I'm wondering if it's possible to get any intermediate values. I've seen a variant of the puzzle before, and I'm thinking it might be possible to light the other end of the rope when one of the shorter rope finishes?
I'm not completely sure if it's possible here though, because I'm given the assumption that "the burn rates in different sections of the rope are unknown and vary", and I'm not fully sure about how to interpret this