Question 12
For each subset $T$ of $U = \{1, 2, 3, \ldots, 18\},$ let $s(T)$ be the sum of the elements of $T,$ with $s(\emptyset)$ defined to be $0.$ If $T$ is chosen at random among all subsets of $U,$ the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m.$