Positive integers $a, b, c$ have greatest common divisor $1$. The triplet $(a, b, c)$ may be altered into another triplet such that in each step one of the numbers in the actual triplet is increased or decreased by an integer multiple of another element of the triplet. Prove that the triplet $(1,0,0)$ can be obtained in at most $5$ steps.
Problem
Source: 2012 Romania JBMO TST 3. 3 , KoMaL Competition, 1999
Tags: combinatorics, number theory