How many subsets S of {1, 2, 3, ..., 15} contain no two consecutive…

We use a transfer/generating approach by residues.

Process elements 1..15 left to right, maintaining a state = (whether the previous integer was selected, current sum mod 3).

A valid subset never selects two consecutive integers, so after selecting element i we forbid selecting i+1.

Define dp over positions with states indexed by (last_taken in {0,1}, residue in {0,1,2}).

For each integer i with residue r_i = i mod 3, we either skip it (last_taken becomes 0, residue unchanged) or take it only if the previous wasn't taken (last_taken becomes 1, residue += r_i mod 3).

Running this dynamic program across all 15 elements and summing the counts ending in residue 0 (over both last_taken values) gives the number of admissible subsets.

The computation yields 536.

The brute-force enumeration over all non-consecutive subsets confirms exactly 536 have sum divisible by 3.

Final answer536