Given $a$ and $b$ I am trying to find an informal proof that the divisors that $a$ and $b$ have in common are the divisors of the number $n$ such that $n = \gcd(a,b)$.
I think it is obvious that the set of divisors between $a$ and $b$ (let us call it $S$) can't have a number greater than $n$. And therefore $S$ should contain the set of divisors for $n$.
But what I can't figure out is: how do I know there aren't other numbers in between $1 ... n$ that don't divide $n$ but are in $S$? Can I have a hint please?