14. The set $S$ is given by $S=\{1,2,3,4,5,6\}$. A non-empty subset $T$ of $S$ has the property that it contains no pair of integers that share a common factor other than 1 . How many distinct possibilities are there for $T$ ?
14. $\mathbf{0 2 7}$ There are six subsets consisting of one element:
$$
\{1\},\{2\},\{3\},\{4\},\{5\},\{6\} \text {. }
$$

There are eleven subsets consisting of two elements:
$$
\{1,2\},\{1,3\},\{1,4\},\{1,5\},\{1,6\} ;\{2,3\},\{2,5\} ;\{3,4\},\{3,5\}:\{4,5\} ;\{5,6\} \text {. }
$$

There are eight subsets consisting of three elements:
$$
\{1,2,3\},\{1,2,5\} ;\{1,3,4\},\{1,3,5\} ;\{1,4,5\} ;\{1,5,6\} ;\{2,3,5\} ;\{3,4,5\} \text {. }
$$

There are two subsets consisting of four elements:
$$
\{1,2,3,5\} ;\{1,3,4,5\} \text {. }
$$

Hence the answer is $6+11+8+2=27$.