p-adic numbers and GCD

Question

Given two numbers $a,b \in \mathbb{Z}$, how do we prove that the $p$-adic number of $\gcd(a,b)$ is the same as the minimum for the $p$-adic number of $a$ and the $p$-adic number of $b$? Does this involve the Euclidean Algorithm at all?

Answer 1

Perhaps one could argue this using the Euclidean algorithm (I don't see how), but I would recommend proving it using the Fundamental Theorem of Arithmetic. First show that if $p^r \mid \gcd(a,b)$, then $p^r \mid a$ and $p^r \mid b$. Then show the converse of this statement. Now use that the $\gcd(a,b)$ is the greatest common divisor.