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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.

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    It sounds like I would be much more concerned about the relation of $a$ and $b$ to $d$. Let's say $p$ and $q$ are distinct primes coprime to $d$ and $a = pd$ and $b = qd$. I wouldn't care too much about what $d$ is as long as $d \neq 0$. But I'm just running my mouth on a hypothetical with hardly any context. – 2017-02-23
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    No prime factorization is required when using https://en.wikipedia.org/wiki/Euclidean_algorithm. – 2017-02-23

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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.