I was trying to prove that if $d \mid a$ and $d \mid b$ then $d \mid \gcd(a,b)$ but wanted a proof that didn't require me to know that $\gcd(a,b) = ax + by$, i.e. that didn't require me to know that the $\gcd(a,b)$ was a linear combination of $a$ and $b$. With that knowledge the proof is trivial (plus for further understanding I wanted a different perspective on this fact that seems to me that doesn't necessarily require that knowledge).
I was trying to think in terms of factorizations to see if I could make it work and these are my current thoughts: if some prime number $p$ divides $a$, and it divides $b$, then it must also divide the GCD. This seems intuitive for primes but if the divisor $d$ is not a prime, I am not sure how to extend the argument. Anyone have any ideas?