1. The problem statement, all variables and given/known data Prove: If a, b are nonzero elements in a PID, then there are elements s, t in the domain such that sa + tb = g.c.d.(a,b). 2. Relevant equations g.c.d.(a,b) = sa + tb if sa + tb is an element of the domain such that, (i) (sa + tb)|a and (sa + tb)|b and (ii) If f|a and f|b then f|(sa + tb) 3. The attempt at a solution Since a, b, s, t, are all elements of the PID, so is sa + tb by properties of rings. I also know that if f|a and f|b then a = xf and b = yf. So f|(sa + tb) can be written as f|(sx + ty)f, which shows that f|(sa + tb) as desired. I'm just not sure how to satisfy criteria (i) of the definition of g.c.d. I know that if d and d' are g.c.d.'s of a and b, then d and d' are associates, but I'm not sure how to use this to my advantage. Any suggestions would be appreciated!