Problem

Source: First JBMO TST of France 2020, Problem 4

Tags: number theory



Let $a_0, a_1,...$ be a sequence of non-negative integers and $b_0, b_1,... $ be a sequence of non-negative integers defined by the following rule: $b_i=gcd(a_i, a_{i+1})$ for every $i=>0$ Is it possible every positive integer to occur exactly once in the sequence $b_0, b_1,... $