2017/18 AHSMC Part 1 Problem 13

If the elements of $S$ are written in base $3$ then the conditions of the problem translate into, if $\overline{a_1 a_2 \cdots a_n}_{(3)} \in S$, then $\overline{a_1 a_2 \cdots a_n 0}_{(3)} \in S$ and $\overline{a_1 a_2 \cdots a_n 1}_{(3)} \in S$. Hence $S$ should contain all the numbers which can be written in base $3$ only by using $0$ or $1$. Since $2 \cdot 3^6 < 2017 < 3^7$ we conclude that the largest element in $S$ that is less than $2017$ is $1111111_{(3)}$. There will be a total of $2^7 = 128$ numbers in $S$ having the requested property. The answer is (B).