abstract algebra - Motivation behind the definition of GCD and LCM

According to me, I can find the GCD of two integers (say $a$ and $b$) by finding all the common factors of them, and then finding the maximum of all these common factors. This also justifies the terminology greatest common divisor.

However, the general definition used is that $d$ is said to be a GCD of $a$ and $b$ if

1. $d$ divides both $a$ and $b$; and,


2. If $d'$ also divides both $a$ and $b$, then $d'$ divides $d$.

My question is that why do we usually accept the second definition over the first. To me the first one seems very intuitive and simple, and does justice to the terminology. The same query goes for LCM as well.

Looking forward to your response. Thank you!