Let \(a\), \(b\), \(c\), \(d\) be positive integers such that \(ad \neq bc\) and \(\gcd(a,b,c,d) = 1\). Let \(S\) be the set of values attained by \(\gcd(an+b,cn+d)\) as \(n\) runs through the positive integers. Show that \(S\) is the set of all positive divisors of some positive integer.