Prove that for all integers $r, s$ and $t$, that $\gcd(\gcd(r, s), t) = \gcd(r, \gcd(s, t))$.

Question

Prove that for all integers $r, s$ and $t$, that $\gcd(\gcd(r, s), t) = \gcd(r, \gcd(s, t))$.

I am stuck in this proof. I have tried using Bézout's Lemma but I have no idea how to proceed further.

Any help would be appreciated.

Answer 1

You only have to prove that, denoting $\;D(a_1,,\dots,a_n)$ the set of divisors common to $a_1,\dots, n$, one has $$D(\gcd(r,s),t)=D(r,\gcd(s,t))=D(r,s,t).$$