BALTIC-WAY 2018 Problem 19

An infinite set $B$ consisting of positive integers has the following property. For each $a, b \in B$ with $a>b$ the number $\frac{a-b}{(a, b)}$ belongs to $B$. Prove that $B$ contains all positive integers. Here $(a, b)$ is the greatest common divisor of numbers $a$ and $b$.