Let a b c be integers so that ab c and b c 1 Show that a b

Let a, b, c be integers so that a(b + c) and (b, c) = 1. Show that (a, b) = 1 = (a, c).

Solution

If gcd (b,c)=1, then there exist integers s and t such that bs +ct= 1. Also, a|(b+c), so there exists an integer k such that (b+c) = ak. Then c =ak-b and, therefore, bs + ct = bs +(ak-b)t =1 or, bs –bt +akt = 1 or, b(s-t) +a(kt) = 1. Since s-t and kt are integers, this implies that gcd(a,b) = 1.

Similarly, since bs+ct = 1, and since b = ak-c hence (ak-c)s +ct = 1 or,( ct-cs) +aks = 1 or, c(t-s)+a(ks)=1. Again, since t-s and ks are integers, hence gcd(a,c) = 1.