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sequence of moves changes pair of integers ( x,y) to (x+t, y-s)

Source: 1998 Belarus TST 1.3

December 25, 2020
number theorycombinatorics

Problem Statement

Let s,ts,t be given nonzero integers, (x,y)(x,y) be any (ordered) pair of integers. A sequence of moves is performed as follows: per move (x,y)(x,y) changes to (x+t,yāˆ’s)(x+t, y-s). The pair (x,y) is said to be good if after some (may be, zero) number of moves described a pair of integers arises that are not relatively prime. a) Determine whether (s,t)(s,t) is itself a good pair; bj Prove that for any nonzero ss and tt there is a pair (x,y)(x,y) which is not good.
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