Let set $S = \{a_1, a_2, a_3, \ldots, a_{12}\}$, where all 12 elements are distinct. We wish to form subsets, each of which contains one or more of the elements of set $S$ (including the possibility of using all the elements of $S$). The only restriction is that the subscript of each element in a specific set must be an integer multiple of the smallest subscript in that set. For example, $\{a_2, a_4, a_6\}$ is an acceptable set, as is $\{a_6\}.$ How many such sets can be formed?
My approach to this problem was to start off with using a1 as the lowest subscript, leading to 2^11 options for the rest of the subsets that use a1. I then repeated the process up to a6, and added 1 for the rest of the subsets a7 through a10. What am I doing wrong?
Your value of $2^{11}$ for sets including $a_1$ is correct. Then for sets not including $a_1$ but including $a_2$ you have $2^5$ choices as there are $5$ numbers divisible by $2$ not including $2$. For sets with smallest element $a_3$ you have $2^3$ choices, for sets with smallest element $a_4$ you have $2^2$ choices, for sets with smallest element $a_5,a_6$ you have two choices, and for smallest element $a_7$ through $a_{12}$ you have one. The total is then $$2^{11}+2^5+2^3+2^2+2\cdot 2 +6=2102$$