Number theory: how to prove $\gcd(a,b,c)=\prod p^{\min(a_p,b_p,c_p)}$ $a,b,c\in\mathbb Z\setminus\{0\}$
$\gcd(a,b,c):=\gcd(\gcd(a,b),c)$
How to show that $\gcd(a,b,c)=\prod p^{\min\{a_p,b_p,c_p\}}$?
My idea is to use $\gcd(a,b)=\prod p^{\min\{a_p,b_p\}}$, but I am stuck on it. Any hints?
Thanks!

As an intermediate step prove that
$$
{\rm gcd}(a,b,c)={\rm gcd}({\rm gcd}(a,b),c)
$$
without using the prime decomposition but simply using the literal definition of greatest common divisor.
Then conclude using the fact that $\min(x,y,z)=\min(\min(x,y),z)$.
