Clearly, $n(S)=6 \times 6=36$. Let E be the event that the sum of the numbers on the two faces is divisible by 4 or 6. Then $E=\{(1,3),(1,5),(2,2),(2,4),(2,6),(3,1),(3,3),(3,5),(4,2),(4,4),(5,1),(5,3),(6,2),(6,6)\}$. $\therefore n(E)=14$. Hence, $P(E)=\frac{n(E)}{n(S)}=\frac{14}{36}=\frac{7}{18}$.
Explanation as extracted from the printed page; notation may be imperfect.