Pigeonhole Principle / Number Theory

Question

Let $S$ be a subset of $A=\{1,2,3,...,1000\}$. Find the largest number of elements in $S$ such that for any $a, b \in S$ with $a>b$, $a-b$ does not divide $a+b$.

I've tried numerous approaches, even brute force (listing), but to no avail. Any help would be greatly appreciated :)

Answer 1

Hint. $S$ cannot contain consecutive numbers, because if $a-b=1$ then $a-b$ divides anything. $S$ cannot contain numbers 2 units apart, because if $a-b=2$ then $a$ and $b$ have the same parity, so $a+b$ is even, hence $a-b$ divides $a+b$.

Now, can $S$ contain numbers 3 units apart?