KURSCHAK 2016 Problem 2

Prove that for any finite set $A$ of positive integers, there exists a subset $B$ of $A$ satisfying the following conditions:

• if $b_1,b_2\in B$ are distinct, then neither $b_1$ and $b_2$ nor $b_1+1$ and $b_2+1$ are multiples of each other, and
• for any $a\in A$ , we can find a $b\in B$ such that $a$ divides $b$ or $b+1$ divides $a+1$ .