OBM 2017 Problem 2

Let $n \geq 3$ be an integer. Prove that for all integers $k$ , with $1 \leq k \leq \binom{n}{2}$ , there exists a set $A$ with $n$ distinct positive integer elements such that the set $B = \{\gcd(x, y): x, y \in A, x \neq y \}$ (gotten from the greatest common divisor of all pairs of distinct elements from $A$ ) contains exactly $k$ distinct elements.