Let $p$ be a prime number and $x \in \mathbb{Z}_{0}$. We define the $p$-adic order of $x$ (notation: $ord_{p}(x))$ as the greatest exponent $\alpha \in \mathbb{N}$ so that $p^{\alpha}|x$. Now show that $ord_{p}(ab)=ord_{p}(a)+ord_{p}(b)$ for every $a,b \in \mathbb{Z}$.
Intuitively, this equality seems very obvious to me. However, I don't know how to give a proper proof of it. Can someone give a hint on how to start solving this problem, please? Thank you in advance.
Hint: Start with the factorizations of $a,b$ as $a=p^mz,b=p^nw$ where $z,w$ not divisible by $p.$