For example, let's say you had $a$ and $b$ which are integers and $(a, b) = d$.
Would you write the decompositions for $a$, $b$, and $d$ in terms of the same primes since they have some in common? Or would you first write it for $d$ and then express $a$ and $b$ as multiples of $d$ times some constants, one for $a$ and another for $b$? Just a simple explanation would suffice. Thank you for your time.
Like most mathematical writing, this is a matter of context. If $a=20$ and $b=70$, then we could write: $$a=10\cdot2, \quad b=10\cdot7,$$ or $$a=2^2\cdot5, \quad b=2\cdot5\cdot7,$$ or $$a=2d, \quad b=7d, \quad \text{where }d=2\cdot5,$$ or simply $$a=20, \quad b=70.$$ It depends on what we're trying to highlight by factoring in the first place, so there's no one "right answer" in general.